£. b.C. 1*2 TRANSACTIONS ROYAL SOCIETY OF EDINBURGH TRANSACTIONS OF THE ROYAL SOCIETY OF EDINBURGH. VOL. XLI. EDINBURGH: PUBLISHED BY BOBEET GEANT & SON, 107 PRINCES STREET. AND WILLIAMS & NOBGATE, 14 HENEIETTA STEEET, COVENT GAEDEN, LONDON. MDCCCCVI. No. I. Published February 5, 1904. No. XVIII. II. )> February 16, 1904. >> XIX. III. >» March 3, 1904. >i XX. IV. !) April 9, 1904. )> XXI. V. )) May 4, 1904. >> XXII. VI. )> May 27, 1904. >) XXIII. VII. n July 1, 1904. j) XXIA r . VIII. >j August 5, 1904. » XXV. IX. >) September 2, 1904. >! XXVI. X. >) September 9, 1904. >> XXVII. XI. jj November 19, 1904. )! XXVIII. XII. :> December 14, 1904. ,, XXIX. XIII. >) January 13, 1905. ) J XXX. XIV. )) June 9, 1905. )) XXXI. XV. it March 3, 1905. n XXXII XVI. jl April 15, 1905. ^» XXXIII XVII. ,. April 18 ; 1905. Published May 26, 1905. May 6, 1905. May 13, 1905. June 15, 1905. June 9, 1905. June 30, 1905. July 3, 1905. July 3, 1905. July 31, 1905. July 20, 1905. August 7, 1905. August 30, 1905. September 25, 1905. November S, 1905. November 7, 1905. January 18, 1906. CONTENTS. PART I. (1903-04.) NUMBER I. On Generalised Functions of Legendre and Bessel. By the Rev. F. H Jackson, H. M.S. " Irresistible," .... II. Certain Fundamental Power Series and their Differential Equations, By the Rev. F. H. Jackson, H.M.S. " Irresistible," III. Magnetization and Resistance of Nickel Wire at High Temperatures, By Professor C. G. Knott, D.Sc, IV. The Glacial Deposits of Northern Pembrokeshire. By T. J. Jehu, M.D (Edin.), M.A. (Camb.), F.G.S., Lecturer in Geology at the University of St Andrews. (With a Plate), ..... V. Spectroscopic Observations of the Potation of the Sun. By Dr J. Halm, Assistant Astronomer at the Royal Observatory, and Lecturer in Astronomy at the University, Edinburgh, . . . .89 VI. Theorems relating to a Generalisation of the Bessel- Function. By the Rev. F. H. Jackson, H.M.S. " Irresistible," . . . .105 VII. On Some Points in the Early Development of Motor Nerve Trunks and Myotomes in Lepidosiren paradoxa (Fitz.). By J. Graham Kerr, Regius Professor of Zoology in the University of Glasgow, . . 119 VIII. An Analytical Theory of the Equilibrium of an Isotropic Elastic Plate. By John Dougall, M.A., . . . . . .129 29 39 53 PART II. (1904-05.) IX. On the Measurement of Stress by Thermal Methods, ivith an Account of some Experiments on the Influence of Stress on the Thermal Expan- sion of Metals. By E. G. Coker, M.A. (Cantab.), D.Sc. (Edin.), F.R.S.E., Assistant Professor of Civil Engineering, M'Gill University, Montreal. (With Two Plates), ....... 229 vi CONTENTS. PAGE X. On the Spectrum of Nova Persei and the Structure of its Bands, as photographed at Glasgow. By L. Becker, Ph.D., Professor of Astronomy in the University of Glasgow. (With Three Plates), . 251 XI. The Histology of the Blood of the Larva of Lepidosiren paradoxa. Part I. Structure of the Besting and Dividing Corpuscles. By Thomas H. Bryce, M.A., M.D. (With Five Plates), . . .291 XII. The Action of Chloroform upon the Heart and Arteries. By E. A. Schafkr, F.R.S., andH. J. Scharlieb, M.D., C.M.G., . . 311 XIII. Continuants resolvable into Linear Factors. By Thomas Muir, LL.D., 343 XIV. The Igneous Geology of the Bathgate and Linlithgow Hills. By J. D. Falconer, M.A., B.Sc. (With a Map), .... 359 XV. On a New Family and Twelve New Species of Botifera of the Order Bdelloida, collected by the Lake Survey. By James Murray. (With Seven Plates), ........ 367 XVI. The Eliminant of a Set of General Ternary Quadrics. — (Part III.) By Thomas Muir, LL.D., . . . . . .387 XVII. Theorems relating to a Generalization of BesseVs Function. By the Rev. F. H. Jackson, R.N., . . . . . .399 XVIII. On Pennella balcenopterce: a Crustacean, parasitic on a Finner Whale, Balsenoptera musculus. By Sir William Turner, K.C.B., D.C.L., F.R.S. (With Four Plates), ...... 409 XIX. The Histology of the Blood of the Larva of Lepidosiren paradoxa. Part II. — Hcematogenesis. By Thomas H. Bryce, M.A., M.D. (With Four Plates), ........ 435 XX. Supplement to the Lower Devonian Fishes of Gemiinden. By R. H. Traquair, M.D., LL.D., F.R.S. (With Three Plates), . . 469 PART III. (1904-05.) XXI. A further Contribution to the Freshwater Plankton of the Scottish Lochs. By W. West, F.L.S., and G. S. West, M.A., F.L.S. (With Seven Plates), . . . . . . . .477 XXII. The Nudibranchiata of the Scottish National Antarctic Expedition. I'.v Sir Charles Eliot, K.C.M.G., . . . . .519 CONTENTS. Vll NUMBER PAGE XXIII. On the Internal Structure of Sigillaria elegans of Brongniart's " Histoire des vegetaux fossiles." By Robert Kidston, F.R.S.L. & E., F.G.S. (With Three Plates), . . . .' .533 XXIV. On the Structure of the Series of Line- and Band-Spectra. By J. Halm, Ph.D., ....... 551 XXV. On the Hydrodynamical Theory of Seiches. By Professor Chrystal. With a Bibliographical Sketch, . . . . .599 XXVI. On a Group of Linear Differential Equations of the 2nd Order, including Professor ChrystaVs Seiche- equations. By J. Halm, Ph.D., ........ 651 XXVII. The Tardigrada of the Scottish Lochs. By James Murray. (With Four Plates), ....... 677 XXVIII. The Plant Remains in the Scottish Peat Mosses. Part I. — The Scottish Southern Uplands. By Francis J. Lewis, F.L.S. (With Six Plates), ....... 699 XXIX. Semi-regular Networks of the Plane in Absolute Geometry. By Duncan M. Y. Sommerville, M.A., B.Sc. (With Twelve Plates),. 725 XXX. A Monograph on the general Morphology of the Myxinoid Fishes, based on a study of Myxine. Part I. — The Anatomy of the Skeleton. By Frank J. Cole, B.Sc. Oxon. (With Three Plates), . 749 XXXI. The Life-History of Xenopus lsevis, Daud. By Edward J. Bles, B.A., B.Sc, Assistant in Zoology at the University of Glasgow. (With Four Plates), . . . . . .789 XXXII. Calculation of the Periods and Nodes of Lochs Earn and Treig, from the Bathymetric Data of the Scottish Lake Survey. By Professor Chrystal and Ernest Maclagan-Wedderburn, M.A. (With Two Maps), ....... 823 XXXIII. The Alcyonarians of the Scottish National Antarctic Expedition. By Professor J. Arthur Thomson, M.A., and Mr James Ritchie, M.A. (With Two Plates), . . . . . .851 Appendix — The Council of the Society, . . . . . .865 Alphabetical List of the Ordinary Fellows, . . . .867 List of Honorary Fellows, . . . . . .886 List of Ordinary and Honorary Felloivs Elected during Session 1904- 1905, ........ 888 viii CONTENTS. 41 f£B.19D7 PAGK Fellows Deceased, 1904-05, . . . . . .889 Laws of the Society, ....... 893 The Keith, Mahdougall- Brisbane, Neill, and Gunning Victoria Jubilee Prizes, 899 Awards of the Keith, Makdougall-Brisbane, and Neill Prizes from 1827 to 1904, and of the Gunning Victoria Jubilee Prize from 1884 to 1904, 002 Proceedings of the Statutory General Meeting, 1904, . . .911 Index, .......... 913 ". tf? TRANSACTIONS OF THE ROYAL SOCIETY OF EDINBURGH. VOLUME XLI. PART I.— FOR THE SESSION 1903-4. CONTENTS. I. On Generalised Functions of Legendre and Bessel. By the Rev. F. H. Jackson, H.M.S. "Irresistible," .......... (Issued separately 5th February 1904-) II. Certain Fundamental Power Series and their Differential Equation*. By the Rev. F. H. Jackson, H.M.S. " Irresistible," ....... (Issued separately 1 6th February 190 %.) III. Magnetization and Resistance of Nickel Wire at High Temperatures. By Prof. C. G. Knott, D.Sc, .......... (Issued separately 3rd March 1904.) IV. The Glacial Deposits of Northern Pembrokeshire. By T. J. Jehu, M.D. (Edin.), M.A. (Camb.), F.G.S., Lecturer in Geology at the University of St Andrews. (With a Plate), (Issued separately 9th April 1901/..) V. Spectroscopic Observations of the Notation of tlie Sun. By Dr J. Halm, Assistant Astronomer at the Royal Observatory, and Lecturer in Astronomy at the University, Edinburgh, (Issued separately Jftli May 190 If.) VI. Theorems relating to a Generalisation of the Bessel-Function By the Rev. F. H. Jackson, H.M.S. "Irresistible," ......... (Issued separately 27th May 1904.) VII. On Some Points in the Early Development of Motor Nerve Trunks and, Myotomes in Lepidosiren paradoxa (Fitz.). By J. Graham Kerr, Regius Professor of Zoology in the University of Glasgow, ........ {Issued separately 1st July 190 %.) VIII An Analytical Theory of the Equilibrium of an Isotropic Elastic Plate. By John Douoall, M.A., ......... {Issued separately 5th August 1904.) Pace 29 39 53 S9 105 119 129 ERRATUM. Vol. XLI., Part I., No. VI., p. 105, line 9 from the bottom of page, "multiply - by p***" 1 ' -*■ (M) EDINBURGH: PUBLLSHED BY ROBERT GRANT & SON. 107 PRINCES STREET AND WILLIAMS & NORGATE, 14 HENRIETTA STREET, CO VENT GARDEN, LONDON. MDCCCCIV. Price Twenty- one Shillings. TEANSACTIONS. I.— On Generalised Functions of Legendre and Bessel. By the Rev. P. H. Jackson, H.M.8. " Irresistible." Communicated by Dr W. Peddie. (Part I. received July 27 ; Part II. August 6, 1903. Read November 16, 1903. Issued separately February 5, 1904.) Part I. 1. In this paper some properties of the functions J [n] (x , X) , P [n] (a; , X) , Q in] (x , X) will be investigated. These functions are generalised forms of Bessel's and Legendre's Functions. Two interesting expressions are obtained for the sum of the coefficients of x in the series P M (^) and ( ^ ] . Throughout the paper [n] denotes ±- . J [n] (x , X) denotes the convergent series X«+2r~[n+2rJ in which Up- 1 we see that P [n] (xX) denotes Z [,■]![« + r]!(2),(2)„ + , [n] ! represents [1] [2] [3] . . . . [?i] (2)„ „ p+l-p 2 +l -p 3 + l . [n] ! reduces to n ! (2)* „ 2" (1) p n + 1 [2« - 2r]\ Zu K , [r\\[n-r}\[n-2r]\{2) l .{2) n J ;) r.r+2^n-2c a .[«-2cj (2) which when p = 1 reduces to Legendre's function P, Qm](x , X) denotes ^[r]!|> + 2r+l]!(2) r (3) If w be not integral then [n]! must be replaced by the function H p ([n]) which is defined as ■wo - ^Bf.;ii::::::a,/ -^ _j»[»i + 1] [n + 2] [» + 3] TRANS. ROY. SOG. EDIN., VOL. XLI. PART I. (NO. 1). (*) 2 THE REV. F. H. JACKSON ON reducing when y>= 1 to Gauss's expression for n(n) or T(n + l) I Li? ' 3-4 * „ r(«+l) - JUn +1 . n + 3 . n + s n + K K The infinite product (4) is convergent for all values of n except negative integers : (2)„ will be in general TT /l\,1\ x( P +\y . . . (5) n,,(M) n p (M) for the infinite product H^([»]) is j) 2 -l p*_~l p 6 - 1 y' 2K - 1 p* - 1 i? 2 - 1 ^ 2 -l which is (2) n denoting a convergent infinite product reducing for integral values of n to (/y + l)(p 2 +l) . . . (p n +l). The difference theorem for Il p ([x]) is n p ([#]) = [afjll^fa;— l]) . The multiplication theorem which is the generalisation of T(^T(x + -\ ■ ■ ■ T(.<: + n ~-\ = T(nx) . (27r)'^n^- nx is investigated in another paper. As obtained from the differential equations the series are perfectly general with regard to n, and the question of [n]! or II([n]) only arises in connection with the arbitrary constant multiplier of each series. 2. In a paper on generalised forms of the series of Bessel and Legendre (Proc. Edin. Math. Soc, vol. xxi.), I have shown that if J [n] (x) denote the series I x l " +2] a/" +4] ) I xi ' li+ -[2][2n + 2] + [2][4][2n + 2][2n+4] + ' ' " " J ' " (6) then y = J (n ](^) satisfies a differential equation ptfQj + ! 1 - [n] - [ - n]}x d £ + [»] [ - n]y = iW^ . . (7) (/"-' , d d — — ■ denoting — — - . — dz m ° d(x p ) dx GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 3 Introducing a parameter X we see that (6) and (7) may be stated more generally as _ , \ w X 2 z [ " +2] XV+* 1 I , V ~ \ X + [2][2» + 2] + [2][4][2« + 2][>T+] ' ' ' ' f ' W and ^S + { 1 ~ W -[-»]} ^ + M [ - »]» = *W M (^) ■ 0) A" In the series (8) give to the arbitrary constant A the value or generally X n (2) II(Tr?1V ^ en throughout the paper J [n] (cc,X) will denote the series A." r=co A 2 'a; [ " + "' ] 2,r2U4l . . . l2rl.|2M + 2l r2ra + 2rl ' ' ^ 0) (2)»W! J6?[2][4] • • • [2r],[2n + 2] [2 ? z + 2r] which can be written in the form 2(2) r (2) B+r [7-]![» + r]! ' (11) Since [21 [4] . . . [2r] = t 1 . ti i^ 1 L JL J L J p _ 1 p _j p _ 1 r r £ p-1 p - I p - 1 = (2)rW! and in general (r not integral) (^ + i)'n, 2 ([7-]) = (2),.n,([r]) 3. It may be verified at once that jj [0] (*A)j = AJ ra (af-A) . . . (12) d_ dx and in general that for ^-s- } J aj-wj^X) } = AJ r „ +1] (a^ • X) . . . (13) ar w J w (sX) (2) r (2)„ +r [r]! [» + »■]! ^fH-Jr a) p™[2r] <(2) r (2)„ +r [r]! [» + »•]! A. B + 2r ajp"[ 2 '"J = ^(2).C2) Ml [n + r\[ ' ' ^ since [n + 2r]-[n] = L- 1 _^_-i = p»^_i = v "[2r] p — 1 p-1 p-1 4 THE REV. F. H. JACKSON ON Differentiating the series (14) with regard to x p " we see that the differential coefficient of the first term is zero and we obtain the series V A"-"[2r>- - " , (15) ^(2),.(2) B+r [r]![« + r]! r-a ^n+'Jr/g-DtnJjjrtn+Sr-l] = Z(2) r (2) B+r [r-lj!L« + rJ! = Xa:_Mnl 2(2W2),, +1+r [r]![n + l+r]! = Aa-*»)J [B+1] (ajPX) which establishes ^•")^ H " J| " l(a ' A) ^ = Xx ~ vl " ]J ^^"^ • • ' (16) By a change of the independent variable this may be written in the form from which we have by repeating the operations A „ +1) ,_ / ,« + . t „ +11J|)|+ii( ^. + . A) = 1 _A J_L '' J . 1 rf ( 1 rfJ„(a:A) | ) »i» .w ;J " +1 ,/(,/") I .^"^' ( /(,- J "--') 1 f a- 3 d(x"') \ *" dx U" ' i ] ] [ ) a theorem analogous to 4. Similarly we may easily verify that \x"J U)] (x"'-\) = J- I xJ (1] (^A) } . . . (19) and in general that X,^J ln . Yi (x^ + \) = |- | ^J M (o* n A) j- . . (20) from which by repeating the operation lii the case p = 1, this becomes ,V « ',-U„ J x k) = (2 +1 | .r«+'J, 1+ .,(*,A) } GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 5. A very symmetrical differential equation is To prove this we must first show that J t «-«(^)-J[» + i](^) = ^?U^P) • • ■ (23) The coefficient of x pln+2r ' 1] in J^. v (af\) - J^aA) is \n+2r-l \n+2r-l [ r ]! [ n + r- 1]! (2),(2)„ + ,_ 1 [r - 1]! [n + r]\ (2) r _ 1 (2)„ + which is k n+2r . p-'{2n] ~ [r]![« + r]!(2) r (2), !+ ,A The r + 1 th term of the resulting series is therefore [2W] \n+2r p n+ 2r xV[ n+2r-ll which since l (24) a; p[H+2r-ll _ _* _ jgCn+Zr] a; shows us that the series is [ |ij M (.>-,w - ■ • W and we have established the relation (23) between three successive functions, reducing when p = 1 to J„- 1 -J„ + i = -J, .... (26) X Now it has been shown that Ax-''J [ , 1 _ 11 (^ +, A) = ^|.^J [ „ ] (^''A)| . . . (27) Changing the independent variable x to x p " we obtain Xx^'^J^^X) = -^ { ^-"i<'U [H] (*A) } . . . (28) which is W^) =-^""^7^) { ^ -B[ %^) } • • ( 29 > as can easily be verified independently. We have already shown that w**) = x*' 4 'V(x>) \ xA ~ n ' J ^ x ^ \ ■ • - (30 > 6 THE REV. F. H. JACKSON ON Taking (30) from (29) we obtain by (23) that The expression on the right is equal to 4 C T*-C*> -X!^d(^ {'•**)} ■ ' ■ <3l> Now is equal to \p n x im ' and therefore we have finally [ ^ { *«("*) " Jm(* ' V) } = <^«W**) - ^' n ^fm(^) ■ ( 32 > a very symmetrical equation. From (23) it follows that j ln _^x) = _L | [a»jr.(«xp) +i>-«[> + 4] j, 1+ ,(.^) + } . (33> 6. The Function Y m (x\) . If P ( „](#) denote the series A J ^M - MD'" 1 ] „3,.[»-2] . [nJ[n-l]|>-2]|>-3] „ [a _ 4] _ .... I (U \ Then * P[»](#) satisfies the differential equation **& - 3 + { x - w - 1 - » - 1] } * d £ + w [ - » - 1]// = pU-) - p Uo ( 35 > Introducing a constant parameter X we have more generally the series I [2][2»-l] i [2][4][2n-l][2n-2]r i K ' Satisfying ■Of - p S + j i - m- [-■ - 1] } 4E +'w[— >i" - p u*a> - n,-,,c*) <37> Give to the arbitrary constant A the value [2n]i A" (2)»[»]l [«]! * Proc. /Jdtn. Ma(/i. Soc, vol. xxi. GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. Then throughout subsequent work the function P w (a;,A) will denote |< - ')' [» - rJI \J-Xm (2)„_,(2)/ ^-'^-" ■ m J* 1 (x\) denotes the series r=co T9i» — Strll ZC - IV" LIZ ILL \ «-2»'-2„[«-2r-2] / OQ\ J ; [«-r]![r,-2r-2]![r]!(2) n _,.(2) r A X " ' < 39 > We proceed to show that d ■ P [w+1] (a\) _ dP c n x] (g\) = ^r 2a + 1 ]p (^,Ap) . . (40) (/a: aa; js" In P 1)l+ i](xX) +fP [n _i](a;X) the terms involving aj [n_8r+1] give us OT r.r+2/ _ 1 \t [2n- 2r + 2]! yn+l-ir^n+l-irl * ^ '[n-r + l]l[»-ar+l]![r]I(2) r <2Wi-r i ,.„r-l.r+l/ _ 1 \r-l [2w - -li'J. \«+l-2r,-[m-2r+l] ^ l ^ [»-r]![«-2r+l]![/-l]!(2) r _ 1 (2) 7l _ 2 , which gives us an expression for the coefficient of x [n ~ 2r+1] , viz., (-1 ]V ^ [2n-2r]l I [2,;-2r + l] ,_,, [2r] ) ^ J [re-r]![«-2r]![r]!(2),(2)„_, 1 [n-2r+l] ' [ra-2r4l]J 2r+l The part within the large bracket is p -la-2r+l_ I - K j,-l-*r(j]pr - I) (2)- l)[n-2r+l] Putting k= —p this reduces to ^ -2,, [2« + 1] [»-2r+ 1] Therefore [2n-2r]\p- ir [27i+l] _ X[2w+1] ^-, , _ , y - r+2 [2w-2?']! xnSr+lUn-^r+U " 2/ }I [n-r]\[n-2r+l]l[r)\(2) r (2) n _ r p" from which we have at once dP ln+1 jxk)_ dP { „(«*) = W + l]P w (^>p) . . (42) rte etas j?" which leads to P n dVl f* X) = V { O - l]P n -i(a: p ,Xp) +p\2n - 5]P„_ :< (^-Xp) + p°[2n - 9]P n _ 5 (a»Wp) + .... J (43) 8. The relation between three consecutive functions is [n]P w (*A) = X[2 M -l>P [ , 1 _ 1] (^A)- i y'+ 1 [n-l]P [ , i _. 2] (xA) . . (44) which corresponds to the ordinary relation raP„ = (2»-l)a;P B _ 1 -(«-l)P B _ 2 ^Consider A[2n - l>P [tl _ 1 ,(ir*A) - K P [n - 1]P [;1 _ 2] (^A) 8 THE REV. F. H. JACKSON ON The terms involving x l "~ 2r] give us " *< " >r[» - '>'- W \»-,.. 1J , l ,, [ !2^:lji(2),_,, 1 (2),., X ^^' tf ^-''' These may be written _ . s [2» - -V ]! ___ I ^"-^ - j>»-- - p"- + 1 - Kjr^p'-*- 1 - 2 r>- -j)'- 1 + 1 ) ^ I " [» r- r]l [» - 2r]l [r]!(2)^ r (2)r « ~ (p- l) 2 [2n-2r- 1] ] Putting *=//' the large bracket reduces to (p"-i)(y"--'-'-i) (p-l) a [2ra-2r-l] which is w so that the coefficient of x {n ' 2r] is ( - 1)V*"M [2»-2r]l 2r and the series is [nJPwCaA) establishing [»]P,„ l (asA> = A[2w-l>P l „_ 1) (^'A)-p' ! + 1 [w-l]P [ „_, ] (crA) . . (45) 9. Another property of the function is By means of this we establish that Pm(1-P 2 ~) = ^P t »-i](l-J?b .... (47) and if n be integral Pm(1-P 2 1 = /'" • • . . (48) which corresponds to the theorem that the sum of the coefficients in Legendre's series is equal to unity. The proof of (46) is as follows : p,„(x-x) . p. l)y^-> H , [<t . fJ fc- _*$ , tojrf ™ • <«> therefore JLtevfr*) \ = Z(-l)"/>'--" +2 A"--'" r[tT J"^* l .-. [n-2r]^-^ tf(z p ) < J o [■'•]! [» -r]I[n- 2 r]!(2), ( _,(2), L J Similarly, (f(af)l I o Lr _lj ![w - r+ l]. Ln _2r+2j!(2)„_ r+1 (2) r _ 1 a! GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 9' The coefficient of x [n_2r+1] in the expression XW ^{ P,M(afX) } ~ P Jj\ FU ** X) d{xP) is K }P [r]![»-r]![«-2r]!(2) r (2)„_ r l L J ' [n - r+ 1] [n - 2r + 1] [n - 27-+ 2](p n ~ r+1 + 1) J Now since [n-r+l](p n - r+1 + l) = [2ra-2r + 2] the expression within the large bracket reduces to r , _J"2n-2r+l][2r] [? '- 2r]+i3 [n-2r+l] which is p~ 2r [w] [n+1] [n-2r+l] and we may write «jrf?y**) _ ~^W^ _ r Bl r. + 11 y(_i yyv-** O ~ 2r J ! r [»-2, +1] / 50y XX <*(*) * ^ -L M J^+1JZ^ l )P k [r]l[n-r]l[n-2r + l]l(2) r (2) n _,: 1 ^ Now consider cx? m {x v \) - c 1 P [ „_ 1] (a;A.) The coefficient of a: 1 " -2 ''"*" 11 in this expression is C[ l)l> A [r]\[n-r]\[n-2r]l(2) r (2)„_,. ll ' J [r - 1]! [n - r]\ [» - 2r + 1]! (2) r _ 1 (2)„_ r which may be written + [r]![«-r]![»--2r]!(2) r (2) n _ r 1 ' r X? T V' [»-2r+l] r-i. r- ..], [„ _ 2r ], (2) r (2)„_„ 1 y X> + V [» - 2»" + K Tf now c be chosen as and c x the large bracket reduces to X«[n] \p[n] [»][» + !] [n-2r+l] and we have ^'W*) -^»]P M W - §M [-+ 'K*""» [r] , [. _ r] , gViff'l]! (2),(2)„_f "" < M> and this series has been shown in (50) to be i X_2 T m d Vw(x p X) dF [n] (xl>\) = X>[n] { asP^X) -ZFtn-ipty} .... (52) TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 1). 2 10 THE REV. F. II. JACKSON ON We see that if X = p h and X = 1 the left side of the above expression becomes identically zero. Therefore when A =j>- and x= 1 x-[»] j,iy ,,(,-A)- ^Ptn-^X) | must also be zero, which is P M (l-#)=i*P [n _ y (l-2>i) . . . (53) from which we have for integral values of n P w (l-i>i)=jP*P ra (l-i*) ; ; ; ; ; • • • • < 54 ) P w (l-i>i)=i>*P [w _ 1] (l-i'4) and so taking the product of the two sides of the equations P M (l-^)=^Po(l-^)=^ • • • (55) which is [»]! [«]! (2)„ 1 P [2] [m - 1 ] + 1 [2] [4] [2n - 1] [2n - 3] J * < 00 ' that is 1 ■« + = W'W',^ . . . (57) [2JL2?i-lJ [2ra]! v 7 the general term of the series being p .r+i W [ra - 1] [n - 2] . . . . [n-2r + l] [2] [4] . . [2?-] . [2n-l] . . . [2«-2r+l] If we put p = 1 this reduces to a well-known series , n • n - \ n ■ n - 1 • n - 2 n- 3 w ! n ! 2" ~ 2 • 2m - 1 + 2 • 4 • 2n - 1 • 2n - 3 ~ ' ' " = ~~2toT 10. xhe series ;/ . A |^_ / ,,[,H[ :: , i -i]^,..- ! , + ....| . . m is a solution of a differential equation of the form /'■''3'-A + {l-[ re _ v ]_[_ ra _ v _l] }«g + [ii- V ] [-»-»- l>-/(«)-/(«0 (59) for, assuming that y can be expressed as a convergent or finite series of the form y = £A^ GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 11 performing the operations indicated on the left side of the differential equation we obtain {p[m^\ [wij - 1] + {1 - [re - v] - [ - re - v - 1]}K] + [re - v] [ - n - v - l]}A 1 a*" a - A.- 2 [m 1 ][m 1 - lJA^™'" 2 ! + similar expressions in m 2 ™ 3 (60) Since p[ m i] [ n h - ! ] = ["h? - ['"J the expression which is the coefficient of A^ 1 ™ 11 in (60) reduces to {H-[»-"]}{W-[»-"-.i]}A 1 so that we have altogether the series {[toj] - [»-v]}{[i»J - [- re- v - 1]} A^"" 1 - X-^toJ [«ij - ljA^" 1 '- 23 + {[m 2 ]-|>-v]}{[> 2 ]-[>re-v-l]}A^ . (61> + Choose m 2 = m 1 - 2 m 3 = rei 2 - 2 w r+1 = m,. - 2 Also choose {[m r+1 ] - [re - v]}{[«t,. +1 ] - [ - n- v - 1]}A,. +1 = X.- 2 [m r ] [m r . - l]A r Let w x also be so chosen that the coefficient of x [ntil may vanish, then m 1 =n — v or -re — v — 1 For the value m x =n-v m r+1 = n-v-2r we have a _ a / 4 * 41 [«-v-2r+2][»-v-2r+l] ^ A r+1 --A r ^ r [2r][2»-2r + l] " " ( ' and for the value m x = - re - v - 1 we have m r+1 = -re-v-1 — 2r a _ a t [re + v + 2r-l][re + v + 2r] A- + i-A r ^ 2 [ 2r ][2re + 2r+l] ' ' ' ^ 6) From relation (62) we have the series »- A I ^-* 4 =Sftfe^ 3 ® , <-" + i • (64) a solution of the differential equation ^g_X-^+{l-[re-v]-[-ra-v-l]^g + [re-v][-re-v-l>=/( a; )-/K) (65) f(x) denoting the function [»-,][„-„- 1] a { ^-^ - A n - ^^".7] " 3] £*--« + } ( 66 > 12 THE REV. F. H. JACKSON ON We also have from relation (63) ,- A {,>— " + y >+]^"_±J± 2 W---^ + } . (67) a solution of the differential equation in which f(x) denotes [« + y+l][n + v + 2] A j , „, 3 [» + v + 3] [n + v + 4 ] x . a , tJ _„.,. s] I (6g) 7^^^ A V + ' [2][2n + 3] i Since \ is quite arbitrary, replace it by \p v and we then have the series y = const. | A"-^"-J - [ " ~3& ~J~ l \ *\»-*-W-*-* + ... } . (69) a solution of ^iS "W> D " + { * -[»-"]-[-«-"- 1] | *| + [« - v] [ - n - v - 1> =/(*) -/(^) (70) The series (69) is when f is integral, and <n d(*p») w The series and differential equation are analogous to one given by Heine (\- x ify--2{v+l).r d - y + (n + v+\)(n-v)y = () . . (72) i(.? z ax of which the primitive is ,-A^ + S^ <rc . . (73) The sum of the series of coefficients in Heine's series (n -v)(n-v-l ) (n - v)(n - v - 1 ){n - v - 2)(n - v - 3) _ 1 " " 2-2« - 1 + ~ 2 • 4 • 2n - 1 • 2n - 3 is shown in ChrystaVs Algehra, Part II. page 185, or 209 (2nd edition), to be v\2n\ ' ' • V ; The analogous theorem in the general series is , _ ,,+■•[" -v][n-v-l] iv+R [n - v\ [n - v - 1] [n - v - 2] [n - v - 3] /( [2][2w-l] +/ ' [2] [4] O - 1] [2n - .8] -••• = (2) n [n]l[n + vll (m (2), [>]![>]! ' ' * * WDj GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 13 Part II. 1. The series , = A{ ^ ' ] -*> 3 L [2 ] [8,-1] J X** + } * • (1) has a general term w iy, ,,+■ ["-"] [?t-v-l] [re-v-2r+l] J_ [ "-"- 2 '' [2]W .... [2r].[2»-l][2n-3] [2n-2r + l] A*f which may be written if n — i> be positive and integral /_ irA . r ^ [«-v]![»]!(2) M [2»-2r]! l a . D -^ fl ^ ' P [2»]! [n - r]! [n - v - 2r]! [r]! (2) f (2)„_, X* so that if we give to the arbitrary constant A the value X"- [2n]l (2) [n]\[n -v]l(2) n we have the series y — "V / _ 1 \r n r.r+2 I™. ILL . \n-»-2r~ln-r-»l V h )P [n-r]\[n-v-2r]\[r]\(2) r (2) n _ r A x which will be denoted by P" («,X) When n and v are not integral [ "i ( ' n<r»])n([» - v]).(2)„ 1 * [2][2«-i] x 2 " + ' • / When p = 1 this function reduces to the function denoted m{nv) by Todhunter {Functions of Laplace, Lame, and Bessel, p. 80). We see that ^' P C > X > = X'O"*) .... (3) Also P; (asX) = A.-' 5' .. ! P M («F.X ) ) . . . (4) P;^k) = x-^%^) .... (5) For brevity P^ will often be used to denote V (x.\). The function P^ satisfies the differential equation d [2) Y i d m V dV or VV 2 ' 1 d(z»- 2 Y 2) dW> I ' w 14 THE REV. F. H. JACKSON ON When p=l the right side of the differential equation vanishes and the whole equa- tion reduces to (x- - \-»)f| + 2(v + 1)'^ - (n - v)(n + v + \)y = ax ax 2. In this article we shall show that 1 _ Jn -v][n-v- 1] ■> ,•■ [» - v] [» - v - 1] [w - v - 2] [» - v - 3] iv _ (7) P [2]t2«-l] P +P ~ [2][4][2n-l][2n-3] P K) of which the general term is +1 [ra - y] [n - v - 1 [n - y - 2r + 1 . ,„ P [2][4J • • • [2r].[2n-l] [Sn-Sr+lf (8) (9) may be expressed as a convergent infinite product U([n])U([n + v]).(2) n H([v])n([2n]) • (2), reducing when n and v are integers to Ml[n-fcy]r.(2). [v]I[2n]I(2), as was anticipated in art. (10), Part I., by analogy with the sum of , n — v • n — v — 1 n — v • n — v — \ • n — v — 2 • n — v-3 " 2 • 2ra - 1 ~ 2 • 4 • 2ra - 1 • 2n^3 _ n\ n + v\2 n -' v\2n\ In the Proceedings of the London Mathematical Society, series 2, vol. L, " Series connected with the Enumeration of Partitions," the following theorem is obtained : — r=oo [P*+v z ] m = [P x z]"' + 2 p "{ ,'" } [P^] m - r (P y )V . . . (10) [ft]- denoting 1 ^ (P ^ 2 + 1 A )(P ^ + 1) ' . . (P^ z + 1) P>1 (j»jy n (P"-l)(P»+'-l) . . . (P^+^-l) o P" 1 ' _ I . P'"-' ! _ 1 -pm-r+ll _ 1 P'-l • P 2 '-l P r '-1 PI I, l , ', I, h » Pi-ptPiPi ■ ■ ■ ' Pi x » Pl X P2 r s l x l I, p x p . . . . p s x 1 x a . . . . x, ViVi ... ■ ft being sets of s independent elements. GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 15 When m is integral and positive 4 [P*2] m reduces to the product of m factors (P x z+l)(F x+t z+l) (P*+'^z+l) Dividing both sides of (10) by [PV] m we have [T>x+y„-\m «•=«= pmi _ 1 p«-r+l! _ J P y - 1 pu+^ll [P*z] m ~ X "*" <£- P - 1 P" - 1 P*+<"-» 2 + 1 . . . . p^+m-r+IJg + 1 V I putting Z = 2 to = 2 a = -1 y = -n+v + 1 a; = » + v + 1 the general term of the series becomes ■pn-r 1 pn— y— Sr+J _ 1 p-n+M-l _ 1 p-n+i>+2r-l _ 1 ( iypr{n+v+l) r — l r i . t - V x / p2 _ -l p2r _ J 1 _ p'.'n-l 1 _ p2n«-2r+l / nT rUr [»-v][»-v-l] [«-v-2r+l] 1 ' [2] [4] . . . [2r] • [2»- 1] . . . . [2n-2,- + l] r which is the r+ 1 th term in, Const, x Pr n ,(l • p*~"). The infinite product L r ^ , when we make the same substitutions as in the series, nPOOTUGS L (P" +y -l)(P" +1 '- 2 -l) . . . (P"+»-ac+" - l).(P"+»-i -1) . . . (p»+-a»+i_l) r-. (P 2 ' - 1)(P 2 '- 2 - 1) . . . (P*^ fc+ *-lj-(P*- 1 -l) . . . (P-'"-^+ 1 -l) " 2 > P>1 which since p2* _ 1 . pSy— 2 _ l p2»— 2it+2 _ 1 = (P"-1)(P"- 1 -1) . . . (P"- K + 1 -1).(P , '+1) . . . (P"- K+, + l) and p2n-l _ i p p2n— 2 _ I p2n— 2k+1 _ 1 _ (P 2 "-1)(P 2 "- 1 -1) .... (ps-ac+i_l) = (P 2 "-l)(P-"- 2 -l) . . . (P-"-^+-_l) we are justified in writing n([» + y])n(M).(2) n a3 v n([2n])n([v])(2), • • ' • by analogy with Gauss's II function. Moreover, when n and v are positive and integral, the infinite product reduces to [> + v]![>]!(2)„ [■2n]l [ v ]\ (2), We have now n([« + vj)n([n])(2)„ 1 _ Jn- v ][n-v-\] 2 (U) n([2»])n([v])(2). ■ p [2][2»-i] p + - ■ K ' 16 THE REV. F. H. JACKSON ON and since rw(3!A) " n([n])II([«-#)„r y . [2][2*-l] A 2 + / putting A = p*-" x= 1 we have If v = this gives P w (l -J>*) = P> .... (16) which was obtained in art. 9, Part I., by another method. p ,» m _ n([n + ^]) (17> When v=± which when p = 1 gives ^ n"(»-j) 1 .I^+i) .... (18) -T« — ; — TT/„, 1 \ x ' 3. Consider rars^iisr -* [B toix + 1 ■ ,19> the general term of the series within the bracket being I : r]l (2)„(2)„_ cW 1 Then if we perform the operations indicated by ■ v ) we obtain a series of which the general term is ^ '[n]r [n-r]l[r]l [»-»-2»«]l (2) 7 .(2)„_ r V ; This reduces to ( ^ [ W _,.] ! [ w _ v _2r]![/-]!(2),.(2) )( _, A viz., the general term of A" + "P [ "„ ] (yP" + ". A) .-?;„(.,-.) - ^d-{^' - ,w^:; + ....} • (22) reducing when p = 1 to 2"A"P„(a-A) = 1 D"+" | A-./-- 1 i" . . . (23) The expression 72)7 - pW (5KT • m GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 17 cannot be expressed as a product of n factors analogous to (X 2 x 2 — l) n , but it possesses the property ^ -^Wt^^t + + (-1)V'" +2 - ! ' (2)„ r L J (2)„_ 1 (2) 1 [r]l[n-r]\ (2) n _ r {2) r _(A 2 -p s )(\ 2 -p r >) .... (A. 2 -p 2 " +1 ) 1 A 2 for z 3 ,, 1C 2 „ I w a positive integer for m 2 >3-X2. J p5_X2 „2n+l _ \ 2 1 P A / _ 1 \n x (25) (2)„ which is a particular case of *[P, x a]»=l + 2 : P re+rj ^ I,z {"K • • • (26) !' = 1 P2<1 substituting r P 3_ 1_ "- l + y ( -ny.^ M[2"-2] ■ • • [2n-2r + 2] 1 L * A 2 J " + <f ( " [2] [4J . . . [2r] W which may be differently expressed (x'-yXA'-p") . . . . (x 2 -^»^) _ *> x»-> (2). " (2) B n "W). * ' If we use the symbol I (n+ " ) as the reverse of D (n+y) we have from (22) I ( " + V (xP n+ " • X) - (^-.P'X* 2 -* 5 ) (A 2 -jr"+i) , 1 J.J* A) A- M ! (2),, • • (28 > jj 2 - 1 • p 4 - 1 . . of which particular cases are r.y - P * =P~and J- PL** -p)=p rirrSwoT • 29 > 1=1 [ ] * =1 [ ] |_»J! L' M J' (2)n.(2)» 4. It has been shown in art. 9, Part L, that if * I A 2 [2][2n + 3] J and/(x) denote ( A 2 2?i + o ; then p ^ ~ hz& + \ i-[»-4-[-»-v-i]}«| + [»-v] [_»-,-%-/(*) -^> If we change A to \p" this differential equation is identical in form with the differential equation satisfied by P [n] (cc,\). * Proc. Lond. Math. Soc, " Series connected with the Enumeration of Partitions," series 2, vol. i. TRANS. EOY. SOC. EDIN., VOL. XLI. PART I. (NO. 1). 3 18 THE REV. F. H. JACKSON ON The general term in the series Qf„(#.A) is [, + i'r]![» + r]!(2)„ + 3 ,._,(_„_,,_„ [r]![2»+2r+l]!(2) r J so that Z* [r]![2n. + 2r+l]!(2) r * A ** Now the general term of the series (30)-:may be written (31) ,,. [2n + 1 1! [n + v + 2r]l [n + r]\ (2)„ +r >y[ _„_ y _. !r _ 1]x _ ar ^ i [« + v]![r]![2n+2r + l]!(2) r ' V ' so that if we give to the arbitrary constant A the value A -„-,_ 1 [n + v]! [2»+l]I and denote the series by Q (x'A) then Stort®^ - Z, [2n + 2r + l]l[r]l(2) r j and ^ = A'/^Q^-A,/) ■ • • (34) If we had chosen Q" [n] (x\) to denote the series which satisfies v-»-l] (33) instead of that which satisfies ^F>y 1 d^y P^^l - r, jJI + etc - M <fo (2) A 2 ^ (2) equation (34) would have taken the form — «egv i = A.^ 2 Q (/I] (^A) Qi-m^j^) satisfies = vj> 3 { qKV • V) - qEV • v 2 ) } ■ • • (35) To find the sum of the coefficients of x in the series Q^-, we make the following substitutions, m= - "J^±l 1 = 2 y = n + v + 2 z= -1 re = - n + v in the series (11). GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 19 The general term becomes prl-n+K) pmZ _ ^ pm-r+li _ | P" — 1 pv+r-ll _ J p rx ! '. i ; — ' — — — ! — '. — ! : : z r P* - 1 P'' - 1 pi+m-lZg _|_ Y m -px+m-rl z + J ■p-n-v-1 _ Y -p-n-v-tr+l _ } pn+"+2 _ J -pn+v+2r _ ^ P 2 - 1 P 2r - 1 1 - P-'"- 1 _ p-2«— 2r— 1> / p2 , [n + v+l][n + v + 2] [?i + v + 2r] .... [2] [4] . . . [2r].[2» + 3] [2n + 2f+l] " ' v ' and the infinite product [Pfe]™ becomes L [v -n- 1] [v-re- 2] [v-re-2/c] (37; «— [2v] [2v - 2] [2v - 4] .... [2v-2 5 .+ 2].[-2n-3] . . . [-2n-2«-l] so that we have this product a , + 2 [rc + y+l]|re + y+2] 4 [re + v + 1] [« + v + 2] [w + v + 3] [to + v + 4] P " [2][2n + 3] iJ " [2][4][2« + 3][2n + 5] n([2» + l]) "^Q^l-j,*) (38) n([>+v])n([W]).(2)/ 5. Now take the differential equation = /(x)-/(^ 2 ) (39) and find a solution in the form of a series proceeding according to ascending powers of x. Assume y = A t a; Cmi] + &.&?** + . . (40) Then performing the operations indicated on the left side of the differential equation (39) we obtain from a term Ax [m] the expression p[m] [m - l]Aaf 2[ "' 1+[ - ] - -^[to] [m - l]A^'"- ,] + {l-[ n -y]-[-n-v- \]}[m]A-x Am ~ 1]+1 + [n-v][n-v-l]Ax™ (41) ■ which is {p[m\ [m - 1] + [to] - [m] [n - v] - [m] [_n-v-I] + [n-v][-»-v- 1] } A.c"" ] - i[»»] [m - 1] Aa***"« which reduces further to {[ m ] _ [„ _ v]}{[m] -[- n -v- l]}AaP« - r-^O] [to, - l]Atf° S[m - 2] . (42) 20 THE KEV. F. H. JACKSON ON So that from the whole series we obtain + 1 [mj - [« - ■'] } (Kl - [ - » - f- 1]}A 2 ^M - -J^K] [m 2 - ljA^" 1 - 2 ' . (43) Choose ?>tj = m 2 - 2 /«., = ;/; 3 - 2 Also choose [mj [m-^ — 1] = so as to get rid of the term -tJtM K " 1]A^ [ '"'- 21 yr"A- Then, in order that the expression may be of the form f(x) —f(x p ' 2 ), the coefficients Aj A 2 A 3 . . . . eto. must be chosen so as to satisfy the relation -^-..K+d [%i-l]A f+I = {[m,]-[n-v]}{[m r ]-[-n-v-l]}A r p~ A" since [ m d["'i _1 ] = ° either m 1 = or ibj = 1 for the value »h = ° and a _ _/-"- 3 x y^ D* - v - 2r + 2] [n + v + 2r - 1] , A " +I - jp=* X * [2,] [2,-1] K - A 1 1 - - [ n - y ][" + v+1 ] T P] + I (44) [2][1] /W " ^ r " 1 p'^ 1 ' [1][2] + i { '/ the general term of the series y being A P r - r --X r [n-v-2r + 2] [n-v] • [n + v+ 1J [ » + r + 2r - 1] ^ (4g) "When n is a positive integer and n — v is even, the series (44) is C. T v [n] (x\) as is evident if we consider that there are - - terms in the series, and so by substituting n — v — r for r we reverse the series. The general term becomes , ,(^)L,„_„ + ,, + ,. v ,-,-,, [2r + 2][2r + 4] . . . [„-v].[n + v+l][n + v + 3] . . . [ 2n - 2r - 1] [m . y _ 2r] W [l]p]...-[»-v-2r> (4 ° ^.-^^^ [ K - y ]^_ v . 2 ]_. [2]^ ,, r+2x „_ y _, \n-v-l][n-v-S] [i^ [2»-2r]! „[n-K-2r] [„_,.]. [„-v-27-]![r]!(2)„_ P (2); GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 21 which is the general term of „- r .„-,-2 [n- v ][n-v-2] .... [21 A * * ? - y -ij Lwry -3]... l[i] - p ;/^) • ( 4§ ) If we had used the value m 1 = 1 m r+l = 2r+l we should have obtained a series satisfying the differential equation (39). The series being . ( m A 2 \n-v- 11 rw + v + 21 m , ) .... y = A^ x™--— L ^LL l x m + | . . (49) the general term being / n >y~ r A ["-v-2»-+l-] [»-v-l]-r» + v+2l \n + v + 2r] {-i.) ,(„-,) f2?-+ll! x (50) If n — v be an odd integer and the series is finite, the number of terms will be n-v+l 2 For r substitute — „ r The general term then becomes A [«-v-l][ w -v-3] . . . . [2] xn . y . 2r [2»-2»-]! xin _ v . m . Ap" " +3 4" >+1 [ B + v] [n + v - 2] [1 [n - r]! [n - v - 2r] ! [r]I (2)„_ r (2Y_ showing that the series is const, x P* ix-\) .... (52) 6. In this and the following articles we shall give examples of the expansion of various functions in infinite series of the generalised Bessel's Functions. The three expansions to be considered are analogous to the following theorems in ordinary Bessel's Functions, J.(«) = 2^Tirr T o( :C ) + 2" +1 ' -n + l ! J i (•'■"> + 2»+ 2 ' • n + 2 \ J ^ x) + ( 53 > S m J (a;) = 2"'} { J m (,-) + jJ m+ ^x)+- m 2 ~J m+i (x)+ } . . (54) 2^7^"! = J «(^) + '2 J «+i( a ') + 2^T2! J "+^ a; )" t " • * ( 55 ) the symbol S m denoting m successive integrations in which no arbitrary constants are introduced (Todhunter's Functions of Laplace, Bessel, and Legendre, art's. 418-422). When [*]» denotes !£=* . ? -^i p^-l p' - 1 p — 1 jr — 1 22 THE REV. F. H. JACKSON ON we have an identity [• + //], = M. + Z^-'" ^ ,y 7 ,,» i '"£-1 M "- M ' (56) Subject to a proper interpretation of [x] m , this expansion holds for all values of m provided the series be convergent, the condition for which is p l >\. If m be integral M_ m denotes = — =— ; . . (57) The theorem in its generality is discussed in a paper on " Series connected with the Enumeration of Partitions," series 2, vol. i., Proc. Lond. Math. Soc. In the following work we require only the simple cases in which m is an integer positive or negative. For y substituting - 2n x 2n + 1r I 2 to r (a positive integer) we have O + 2r- 2n\. = [9n + 2r] r + ^y(»+ i 2r ] [ 2r ~ 2 ] • • • • • ^' 2 ''~^ + ^ x [2« + 2r] r _ s [ - 2n], Now [2n + 2r] r _, is [2»+2r][2» + 2r-2] .... [2n + 2a + 2] = [n + r][n + r -1] .... [n + s+1] ■ S 511 and [-2»] a = [-2»][-2ji-2] [-2re-2s + 2] = (-l)^-'" N -' 2+s [2w][2?i + 2] . . . [2rc + 2s-2] = (_l)*p-**^H[»][„ + l] . . . [„ + ,-l].(g"±t> So that we have (2),[r]! = [ n + r][n + r-l] . . . [?i+l f 2 ^+ r { * In + 2/ ^ [r-.]I[.]l (2) r _ s (2) s LM + ,J " • • • L " + ' S+1J (2) n+s [»][» + 1] . .. [w + t s--l]( 2 ^±£- V.^/n-1 Dividing throughout by [n +r]! (2) n+r [r]! (2),. this reduces 1 1 ' , l [n + r]l(2) n+r ~ [n]l(2) n [r]\(2) r + 2-< " ^-^^[r _ s]! ^j, ( 2 ) r _,( 2 ).[» + *]! (2)„ +s * M[ w+ i], [w+g -i3 (2Wi (58) V^/n-l Dividing throughout again by [r]! (2) r this identity becomes 1 = _i . - + [n + r]J [r]l (2) re+r (2) r [»]! (2)„ [r]! [r]! (2) r (2) r 6f " [ W + S ]![ S ]!(2) K+ ,,(2) S - (2)^ W " + *" 1 • (59) [r-*]![r]!(2) r _,(2) GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 23 From which we can see at once that T / W W* T / »" \\ 2 A»+V n+11 r n (2) n r , _H+1 , x + . ■ (60) the general term being subject to the convergence of the series ; for if out of each term of the series on the right side of (60) we pick out the part involving \ n+2r x [n+2r \ we get a set of r+ 1 terms Xn+2r„[n+2r] f 1 * _ „2 1 U1 ( 2 )» 1 1 [»]| (2)„ * [r]l [r]! (2) r (2) r * [» + 1]! (2)„ +1 L J (.2) 1 (2) H+1 [r - 1]! [r]l (2) r _ x (2) r + } • • • • (61) which by (59) reduces to *" + ^ C " +2] - . . (62) the general term of J [n] (x\), — and theorem (60) is established. A particular case of this is *■**> = wfc^-*flmw M <H>T'-<''" !A > + (63> analogous to J B (s) = j^ Jo(«) + 2^- 6 Jl(a:) + * (64) To investigate a theorem analogous to S m -J (x) = 2'" | j m ( x ) + m j m+2 (x)+ m -™±±J m+i (,- )+ . . . . } . (65) If, in the series on the right, we replace the Bessel's Functions by infinite series and collect the terms together according to powers of x, we find that the terms involving x m+2r f orm an i n fi n ite series 2V+ 2 ' i - - - - + M + 1 I I (66) (m + r\r\ 2"'+ 2 '' 1! m + r + 1! r- 1! 2 m+2r 2! ra + r+2! r- 2! 2 m+2r J which is 2 g ( 1 _ m r (. / 67 v 2 m+2r r! r!|ffi + f.ffl + r -l.f+l 1 m + r + 1 . . . . . . r + 1 1 The series within the bracket is by an extension of the notation of Vandermonde's theorem (Proc. Lond. Math. Soc., vol. xxvi. p. 285), (»")-« - ij-W-m-lH + 2! ^->"-^ r) l + (6 ^ which is (r + r)_ m subject to convergence conditions, viz., 2r + l >0 (r + /•)-»> = (2r)_ m = — — — - — ^— -y m + 2r ■ m + 2r - 1 . . . . 2r + 1 24 THE REV. F. H. JACKSON ON So the re-arrangement gives us an infinite series, of which the general term is x m+ ' 2r 1 2 2r -r!r! X m + 2r - m + 2r- I 2r + l which is the general term of S"\J (.r) ..-...- ( 6 9) S" 1 indicating vi successive integrations. 7. The preceding analysis shows us how to construct the analogous series for the generalised Bessel's Functions. Consider the generalised form of Vandermonde's theorem 5=00 ml 1 m— It i m—s-\-ll 1 [«+*]. = M. + 2><-"-» • v J i J {"** i M™-M • (70) s=l f L ' 1 J L • • • Jf 1 convergent for all values of m if p l >\ : If m be a negative integer, ^~ m = [x + ml][x + m^ll] . . . . [x + l] ' (/ ' In the theorem (70) replace m by —ma, negative integer, I by 2, x by 2r, and y by 2?\ Then 5=oo -2m _ 1 r) -2m-2s+2 _ 1 s(2r+2m+2s)JP 1 ■ • i 7 * ^ 2 - 1 jr'-l x Q]|>-2] [2r-2.+ 2] , [2r + 2m + 2s] .... [2r + 2] v ; Now [2r + 2r]_ TO is [4r]_ m 1 [4r + 2m][4?- + 2m-2] .... [4r + 2] and this may be written (2)gr_ (2) 2 , +J „[2r-fm].[2r + m-l] . . . [2r+l] Dividing (72) throughout by [r]! [r]! (2), (2) r we obtain 1 1 [r]! [r]! (2) r (2) r X [4r + 2m] [4r + 2] = 1: + V(-l)y +s+2r [?n+g-l]!(2) m _ 5 _ 1 ,-„, [r + ro]! [r]! (2) r+m (2) r Z. v ' e [ m - 1]! (2)_ 1 [»]1 [>-«]! (2) s (2) r _ s [r + m + S ]! (2) r+m+s ^*' the analogue of 1 1 = 2 m I 1 _ ™ 1 + I (7A\ 2 2r • r! r! m + 2r . . . . r+l I m + r\ r! 2 m+2r 1! to + ?-+ 1! r- 1! 2 m+2 '' ' ' " ' j ^ ' used in art. 6 in the analysis of the series (65). At this point we define the function J^xX) as the convergent infinite series r=ao g "+-''-l T * w+ ' r s '- 1 .... (75) ^[» + r]![r]!(2) r (2) n+ " r GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 25 Then r=ao [2ra+4r] jfM) = 2 A " +2 '^ [2] ■ • • (76) J[ " A ^[n + r]![r]!(2),.(2), !+1 . j£0*) " P 2 - m M • r.y^v Jfi(*Ap) + • (77) Consider the series I 2 ) (2) m _ 1 (2) 1 of which the general term is ( ^ [,]| (2) m _ 1 (2;. J ^M^P) • (78) If we replace in (77) Jf m] Jf m+2 ] by the infinite series which they represent [2ro+4r ] and collect the terms according to powers of x, then the terms containing x [2] form the infinite series [2m+4r ] s= oo [2»i+4r] X'»+- r ./ ; ~ir- + V ( _ iv A m + 2 ^ m ■ [m + s - 1]! (2) m+s _ t p*+H*. ( 79 ) [m + r]! [r]!(2) m+ ,(2) r £* [m - 1]! (2) m _ 1 [s]! (2)J>-«]1 (2) r l,[m + r + «]l (2) m+ , +s which reduces by (73) to [2m+4r] 1 X"'+ 2r r~P]- [2m + 4r] [2m + 4r - 2] • • • [4r + 2] [r]l [r]! (2),(2),. which is ^pTJgS(a^".X) • • (80) I m denoting the operation reversing d(xP 2m -') I dia***-*) I ' " " ' I *W) ) J i 8. It is well known that 2". re! 2 " + 2 2 - 2! 2 3 • 3! Consider the series J[»M) + Sh*^) + ^/^.•W***) + (82) of which the general term is -^ (2) \rV? ln+r] ( x> ' ^ J [n] (ccX) denoting V y [re + r]![r]!(2),(2)„ (83) (84) Replacing J [n] (x\) J [n+ i](x p X) ... in (82) by infinite series, and collecting the terms accord- ing to powers of x, we have for the terms involving x [n+2 '' ] a group of r+ 1 terms, viz., ^n+2r a Jn+2r] ^n+2r x [n+2r] ( " l) \n + r}\ [r]! (2) r (2) B+r + ( " lr \n + r]![r - lj!(2) P _ 1 (2) )l+r (2) 1 + ... + /_ iy-v'-i A * (85) + "- + ( l ) P [ ra + r]![r-s]!(2),._ s (2)„ +r (2),. V ' TRANS. ROY. SOC. EDIN., VOL. XLI. PART T. (NO. I). 4 26 THE REV. F. H. JACKSON ON which is which is identically zero. The only term which does not vanish is the first term in J [H] (a;X) so that we have identically for all values of n *vM) + ^St. W**) + (87) [»]!(2)„ l " lv ' (2) x [l]! If in 84 we had taken J [n] (xX) as '[re + r]![rj!(2) r (2), H the signs in the terms of (86) would not have been alternately + and — ; (86) would not then be zero but 2(» 2 +l) . • . Or r ~'"+ 1) - - Another expansion x [n] in terms of the P functions is [2n]!X»a™ _ ,,., 3 [2re-3] +rfl [2n-l][2n-7] p (8 g, analogous to 2re!z* 1 D 2re-3 D 2?i - 1 • 2?J - 7^ , = "i. + — "n-2 + H 5 "«-4 + re! re! 2" 2 - 2 • 4 9. Various interesting theorems have been obtained with respect to Bessel's Functions j_ when the variable is not x but *Jx. The analogous theorems for J [n] (xp+ 1 ' A*) are given in the following work. It is well known that iJx ,„\x ^(^)) = (-\) m x— J„ +m (» . . (89) ,7m f " 1 n— m _ £n\ X *UJ*)\ = (ir*~ J»-»( */*) • • • (90) Let D« denote ^^ . -^ .... ^ - ^ .... (91) Then which reduces to (89) w T hen p = 1 B |»] 1_ V A -a; ^-^"iTg "'- 1 . . . (93) « [r]l[n+r]I(2W2W GENERALISED FUNCTIONS OF LEGENDPE AND BESSEL. 27 [2r] _, \ r xP n [2] ' ^[r]![» + r]!(2)«(2) B+r ' (94) Operating on this with D tml we see that the operation reduces to zero all terms for which r<jn, while the result of operating on the general term [2to+2t- ] [m + r]\[n + m + r]\(2) m+r (2) n+m+r ' ' * V°> is \2m + 2r\ \2m + 2r - 21 \2m + 2r - 2m + 21 rTTn " • tTvt " • • • • L F7 r=. [ 2m+2r-2m] V*- [ 2 1 [2] [2] ( X p' l+2m ) —m , (96) [m + r]! [n + m + »•]! (2),.(2) n+m+r which is easily reduced to [2]" A * ^ [r]![ W + m + r]!(2) r (2) B+m+r ' , ' ' ""> viz., the general term of ' ^;rj [rt+ i^-W .... os) . [ 2 ] In a similar way theorem (90) may be generalised We have shown that [»] • P W (*A) = k[2n - ljaiP,,,.^, A.) -p" + \n - l]P [n _ 2] (*A) . . (a) In the same way we can establish i^MQwO"*) = X|> - lJcQ^^X) - [ n - lJOu.-flfoX) . . (0) analogous to the ordinary recurrence formulae. Multiplying (a) by Q [n . 1] (x p ,'X) and (/3) by P [ , l _i ] (x^,X), then subtracting, we obtain [n] { P w (a,X)Q [ „_ 1] ( !B ",\) - y^^P^^X) } = [«-l]{P [n _ 1] (^A)Q [w _ 2] («X)-^+ 1 Q [ „_ 1] (^,\)P [ „_ 2] ( a; A)} . . (y) analogous to »{P«Qn-l - Q,,P„-,} = (» - 0{Pn-lQn-2 " Q-lP.-«} So also multiplying (a) by Q [(l _ 2 ](as,A) and (/3) by jt) n+1 P [w _2]( a; >A)» * nen subtracting, we obtain [n]{P w (a« l X)Q Di _^sB I X) -p 2n+3 Q m (x\)? ln -v(xk)} = A[2n - l].r{P [/i . 1) (.r"A)Q [ „_, ! ( a; .X) - p n+i q in - n (x^k)F [n ^(x,X)} |_w- ij When we put as = 1 we obtain [n]{P M (X)Q tn _ 1] (A) - p" +2 QMVm-»W} = [« - 1] {P[»-n(X)Q [n - 2] (X) - P n+ % n -i^Wm-^)} («) 28 GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. and so by repetition, if n be integral, = Li]{P w Qto]-i> 8 Q w iW = 1 the analogue of 1',,,+uQ.- - Q»+il J „ = n + \ Similarly from (S), putting x = 1 and repeating we obtain P [ , 1+1J (A)Q 1 , l - 1 (V) -j^Qj^XJPp^X) = H s " + fl . (,) Ln+lJ-L»J From the recurrence formulae we can obtain without difficulty -"[2r+l] [n + 1] P [ , t+1 ,(A)P l „ l (X')-iy, +1| (A')P M (A) A theorem analogous to Neumann's expansion of an arbitrary function in a series of Bessel Functions I have given in a paper, L.M.S. Proceedings; also the analogue of Lommel's theorem ^ = {J^«)}" + S{J f (a)}* + 5{J f («)}»+ ad inf. ( 29 ) II. — Certain Fundamental Power Series and their Differential Equations. By the Rev. F. H. Jackson, H.M.S. " Irresistible." Communicated by Dr W. Peddie. (MS. received December 7, 1903. Read January 4, 1904. Issued separately February 16, 1904.) The series which will be discussed in this paper are of the general type Aj**"") + A 2 x l "^ } + +A r x <mr) + . . (1) Consider a sequence of elements Pi,p 2 ^P3, then (w r ) will denote the sum of the first m r elements of the sequence. The simplest connection between the terms of the sequence is equality Pi=P2=lh= =Pk= • • • The series expansions of the ordinary functions of analysis and the series which are solutions of ordinary linear differential equations belong to this simplest type. Another more general type of series is formed when the elements pj,p 2 ,Ps • • • • are m geometrical progression as a, ap, ap 2 , In this case the index (ft) denotes a + ap + ap 2 + + ap n ~ 1 , which is a r , the limitation of n to positive integral values (where the sequence is geometrical) may be removed, as will be seen in the particular discussion of the functions J [n] (\x), P [n] (\x), Q [n] (^x), F([a] [/3] [-yjAa:). lip be made unity, the geometrical progression becomes a progression of equal elements and the properties of the general functions reduce to analogous properties of the simpler functions ~F(aftyx) ; P; t (x) ; Q n (x), etc. Euler's expansion (1 - x)(l - a; 2 )(l - x s ) ad inf. = ^( - l)".rf' 3 " 2+ ' !) (x < 1) and Gauss's series ( l-a;' 2 )(l -a: 4 )(l -x 6 ) . . . ad inf. _ -y a .'^i 1 (1 - x)(l - x 3 )(l - x?) . . . ad inf. ~ ^ (x<\) are particular cases of the general series (l) for the sequences 1, 1, 3, 2, 5 . . . , 2n — 1, n, . . . and 1,2,3,4, . . . . respectively. The fundamental Hypergeometric Series is F- 1 + a l' a 2' a 3 % ' 1 >+ a i( a i + (1) ) a,(a,+ (1)) (2)+ ( a\ MPi-PfPs p t (P(2) y 8 1 (^ 1 + (2)-(l)) .... £(/J 5 +(2)-(l)) ' VW In the case when there are only three elements a /3 j in the Hypergeometric Series and the sequence of p.s is 1, 3, 5, . . . . , 2ft +1, . . . the following series are interesting cases : TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 2). 5 30 THE REV. F. H. JACKSON ON CERTAIN U «i, + a-a + 1 -p-p + 1 . a-a+1 a + (n - l) 2 • /?•/? + 1 /? + (rc-l) 2 , l2 m l 2 y l-.-2-.y.y+3 ' ^•••■■ r - l-.^.3- ;/-yy + 3 . . . . y-l+W- '"' V/ + l 2 y + l l 2 -2 2 .y+l 2 -y + 2 2 W l + a ^ + a . a+ l ! .y + l V + (5) It is well known that if II g (a + m) denote the product of s factors (aj + m)(a. 2 + m)(a s + ra) .... (a„. + to) to integral an identity of the following kind can be established, II 5 (a + to) = B + B x to + B 2 to(to - 1)+ .... +B 5 to(to-1) . . . (to-s+1) where B Bj . . . . B 4 are constants, that is, are independent of m. We proceed to establish a generalisation of this, on which all subsequent work will depend. Let n„(a + (m)) denote the product of .s factors ( a l+Pl+P2+ • • • • +Pm)( a 2 + Pl +P-2 + ■ ■ • +Pm) (a s + p 1 +p.,+ . . . +V m ) then n s (a + (ra)) is identically equal to B u + B (OT > + B (m)((TO) - a)) + . + b (ot)((w) " (1)) ((■>») -(*-!)) _ (6) 1 Ih P\Pl P1P2P3 ■ ■ ■ Ps which may be more conveniently written B + B<^ + B 2 ^ + + B (™h . . . (7) Pi P1P2 Ps*- The coefficients B B 1? etc. are independent of ra and are given by gB„=|<- 1)V ^n(, +(> ,-,,). . . . (8) Before proceeding to obtain these coefficients it will be well to explain the notation clearly. (m — 1) is not the same as (m) — (1), for (ra — 1) denotes P1+.P2+P.3 + . . . +p m -i , while ( ra) - ( 1 ) denotes p. 2 + p 3 + p 4 + . . . +p m (m), = (p 1 +i? 2 + . . . +p m )(p 2 +P s + • • +p m )(p 3 + ■ ■ • +P m ) • • • • (p,+ ■ ■ ■ • +Pm) B s -^ ! n(a + (3))- ( ^ r! n(a +( 2,) + (J J^ J] n(a + (i) ) -Mn W . . (9) (3)! = 0?! +^2+^8)^2 +^3)^8 I 3 }' = (P S +P-2+lh)(P2+Pl)Pl (2)! = ( Pl +p 2 )p 2 {2}! = fa+pjfa (1)!=P, {1)1 -JP. .... (10) In the expression for B 4 * four elements p 1 , y? 2 , p 3 , p± appear and (4)! = (p 1 +ia 2 +p % +p i )(p 2 +p z +pd(p 3 +PM I 4 }' = (Pi +P& +P2 +Pi)(Ps +P2 +Pi)(P2 +Pi)Pi (3)! = ( Pl +p 2 +p 3 )(p 2 +P 3 )p 3 {3} ! = (p A +p 3 +p 2 )(p 3 +p 2 )(p 2 ) (2)! = (p 1 +p 2 )p 2 {2}! = (p i +p 3 )p 3 (1)1 = Pi {1}!=^4 (H> FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 31 The symbol { } ! must not be dissociated from the expression B in which it occurs, because {r\ ! as formed from n elements p x p 2 . . . p n is not the same expression as \r}\ formed from m elements p x p 2 . . . p m . In general, for n elements {1}! =Pn {2}! = (Pn+Pn-^Pn-! {3}! = (f»+j'»-l+P»-!)(ft-l+ft-j)j), l - 2 . ■ • (12) At this point some properties of the coefficients _(«)! (»- may be noted (n-r)\ {r}\ (n)l which is the generalisation of ^ K '(n-r)\{r}\ 2(-iy »! = r\ n - r\ As an example of this property take (p 1 , p 2 , p 3 , p 4 ) = ( 1 , 7 , 9 , 5), then The expression (4) (3) (2) (1) {1} {2} {3} H) (4)! = (l + 7 + 9 + 5)(7 + 9 + 5)(9 + 5)5 = 22-2M4-5 = (l + 7 + 9)(7 + 9)9 = 17-16-9 = (1 + 7)7 = 8-7 = 1 = 5 = (5 + 9)-9 = 14-9 = (5 + 9 + 7)(9 + 7)-7 = 21-167 = (5 + 9 + 7 + l)(9 + 7 + l)(7 + l)-l = 22-17-8-1 (4)! (4)! (4)! (4)! IS (4)!{0|! '" (3)!{1}! T (2)! {2}! " (1)! {3}! T (0)! {4}! , 1078 55 55 735 n 1 — 4- — _ — + = 408 + 12 4 68 If p x p 2 , . . . be an arithmetical sequence 1, 1 +a, 1 + 2a, etc., then Z (»)! (2-K»-l)a)(2 + «a) .... (2 + (2n-2)a) (n-r)\{r}\ ~ (!+«)(! + 2a) .... (l + (M-l)a) a generalisation of •I n — r\ an. (13) (14) to which the identity reduces when the sequence (P1P2P3 • • • ) = (1 > 1 > 1 • If the sequence is (p x p 2 ■ ■ . ) = (1 , 3 , 5 , . . .) '(r)\ {n-r}\ = 2 2 •)• (15) 32 THE REV. F. H. JACKSON ON CERTAIN The identity (14) is a particular case of F(« /3 y 8 e 1) and may be thrown into the form (x)„ = (2.r)„ + 2^^^ -2r)„_ r (2.r -r+1), . . . (16> iii which (.«)„ IS xx-Y-x-l . . . x-n + l A more general form is (17) (m M. = DH. + z ^ f3l g J : r3 |t fa - 2 *W 2 * - '• + ^ • r i _ 1 r -| = P'-I " = M [*- !] [»-» + 1] [ft:J ~ p - 1 These and other interesting theorems due to change of the sequence (p x p 2 must be left to another paper. We now proceed to obtain the coefficients B B x , etc. Suppose that II s (a + (m)) is capable of expansion in the form Etf. + (»))» B + B 1 ^ ) + B 2 ^ + + gM. . Pi i>2 ! A 1 (m) s denoting (^ + . . . +p m )(p 2 + • • • +Pm) . . . . (p s + ... + p m ) PJ- „ Pi'PiPz • • • ■ Ps In (18) substitute (m) = 0, then we have B = n s (a) Similarly, if we substitute (m) — (1) = we obtain b +B i a=n(o+(D) Pi Continue the process of substitution by putting successively (m)-(2) = ( m )_(3) = We obtain the following set of equations for determining the coefficients B B l5 etc. n(a) = B n(a + (l)) = B 0+ ^B 1 Pi U(a + (2)) = B +Pl±l2 B + Pi +P2-P-2 B .... (19> Pi P1P2 n(a + (n)) = B + ( tt )B 1 + ^B 2 + .... + @?B n Pi Pj- P» l FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 33 From these equations we obtain B = n(a) B 1 = n(« + d))-n(a) B PiPi n(g + (2)) -giggn(a +(!))+ PlPa n(a) . .' . (20) Pi + P-?P-2 PlP2 Pl'lh +P\ _ ,j n(a + (3)) n(a + (2)) n(g+(l)) 11(a) I 3 P3 \ (3)!{0}! (2)!{1|! + (1)!{2}1 (0)! {3}! J To establish the form of the coefficient B„ we assume that the law of formation holds for BoBjI^ . . . . B n . v namely, that ,/ n(q + (tt-i)) u( a + (n-2)) , t n(g) i , n Hn-i-Pn-i-\ („_!), {0 }! _ (n"-"2)TpT! ( ' (0)!{«-l}!l ' ( > . ,/ n(a+(»-2)) n(a + ( «-3)) 4.(--i\n-* n(a) i "- 2 ^- 2 '{ (re-2)!{0}! (n-3)!{l}l V ' (0)! {«- 2}! J From (19) n(a+(K)) = B + B 1 ( -!i ) + +B r ( -4'+ • • • • + B«£& Pi ft! ft! Replacing B B 1 . . . . B^^ by the expressions (21) we obtain B„ = A.II(a+(ra)) + A. l II(a+(rc-l))+ + A s II(a + (n - s)) + .... + A„II(a) . (22) where A 1= - ft! (»)»-! 1 (»)!(»- 1)!{0}! X _ P» ! ( OOn-l _ (")«-2 I 2 (n)ll(n-S)I{l}! (rc-2)!{0}!j \ _/ _ ly+li 5 "- / ( w )n-l _ W»-2 ■ i / _ I \s \ n )n-s \ /oo\ 5 v ; (»)n(n-«)l{*-l}~l (n-s)\{s-2}l K '(n-s)[{0}lj ' { ' {s — 1}! in the first term of X s is formed out of the set of elements p^ p 2 . . . . p n -\ and is (ft-l+ft-2 + ■ ■ ■ + Pn-s+l)(Pn-2 + ■ ■ ■ +Pn-s+l) Pn-s+l {s — 2}! in the second term of A s is formed out of the set of elements p 1 p 2 .... p n -z. and is (ft-2 + • • • +Pn-s+l)\Pn-3 + • • • + Pn-s+l) Pn-s+l • We see that \ which is ft! («)«-! (»)!(»-l)!{0}! may be written ft! . ( Pl+ ■ ■ ■ +Pn)(P-2+ • • • • +Pn) • • • (ft-l+ft ) (n-l)l ( Pl + . . . +p n ){ p . 2 + . . . +p n ) . . . (p n ^+p n )p n ft! ft! (n- 1)1 ft (»-l)l{l}l since p n is {l}! for the n elements p x p 2 . . . . p n . 34 THE REV. F. H. JACKSON ON CERTAIN Similarly X 2 may be written PJ / (»)»-! _ ( »)»-g I (w-2)!l (»)!{1}I (»)!{0}!J which is /'« ! f__L_ _ —J— I (m-2)! j yv/)»-, Pn-1+iV.ftJ (* J»n! " (»-2)liWi+lVJ>W-i (»-2)!{2}! since (_p„ _ x +p„)/?„ is {2}! for the n elements PiP 2 . . . . p n . In the expression (23), if we reverse the order of the terms we have and since in K, (n-8)!l(n)!{0}! (n)l{l}! V ' (»)! {*- 1}! ' " V H.-i _ M Vyp = Pn(pn + Pn-i) (p» + P„-l + + /'«-»+l) also {0}! = 1 {1}! = £>„_ s+1 (Because the term involving {1}! was derived from B„_ s+1 ) {2}! = (/V- s+ 2+iV- s +iK_ s+ i {3}! = (p n -s+S+P n -s+2+Pn-s+l)(Pn- S +2+Pn-s+2)P n - S +l {S-1}\ (pn-l + + Pn-s+l)(Pn-2 + - ■ • + Pn-s+l) P«-s+l We see that for the set of elements p n p n . 1 Pn-i+i the expression \ n )>i- s _ ( n )n-s+l , , (_1\J-1 \ n )n-J (n)\{0}\ (n)!{l}I ■*"•••" "^ > (n)I{«-l}l is W " (— 1)!{1}! + +( - ir \l)l{s-l} ■ ■ • (25) but for any set of s elements in any order + + (-1) s t4 = ° so that ® - Fwri + + ( -y*W=m - - ( - 1> 'ft ! • ■ < 26 > and we have A. = (-i)V Vn\ (n-s)\{s}\ which establishes the form of B rt in general b, - |(-i>' ( - i dyrw! n (" +( "- s »> ■ < 27 > FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 35 The General Hypergeometric Series. Let (p denote the differential operator B ft + B 1 aj'"D (I) + B 2 a*'+^D (2) + +B s a; ,s »D (s) . . . (28> and -vf/- denote the operator ^|c x ,1, D |1) + C 1 a; , -- ,, D« 2 »+ + C^'D'" I (29)- B„ being' £(- If ? ^r^i n ^ a + (n " r » ■ ■ ■ ( 30 > r=0 °- § ( - 1) V^W w+< * , - r>) • • <»> Then if y denote a series of the form 2Aa; (m) that is A 1 a; (m ' , + A 9 cc , ' %) + and we operate on y with <p — \J/- we obtain {^ - f }y= T A.n s ( a f (m))x<»" - T A ( ^n^ + (m) - (1))^-'H . . ( 32 ) The lowest index present in the series on the right is (w^ — (l), and the term in which it occurs is A 1 (-iin,(£+ (wij) - (l))a ,m >>- (1 > Pi Choose mj so as to make this term vanish, the possible values of m 1 are zero and (1) — j8. Now, from the manner in which (m x ) (w 2 ) .... are formed, we cannot have any such relation as (m r ) = (m r+1 ) — (l) unless the elements p 1 p 2 p 2 . . . . are equal to one another, so we choose (»» 2 )=< 1 ) = /'i (m 3 ) = (2) =j>! +i> 2 and A Pi A r+ ^r±^-U t (/3 + (m r+1 ) -(!)) = A-ir^q + (m r )) Then we have F = l + », n ^ g ) gin + » 2 n,(a)n,(q + (i)) (2) , 33> 1+Pl (i)iW +A (2)in < 08)n^ + (2)-(i)) a ' + (dd> the general term being » n,(a)II,(q+(l))II,(q + (2)) n s (a+(n-l)) r{n) , ^ (»)!n,C8)n,( j 8 + (2)-(D) n,G8 +(»)-(!)) ' ' and a differential equation 36 THE REV. F. H. JACKSON ON CERTAIN («-♦>'-"*> { o ^ flffft?! : : : : : %? J ^ ■•■•) -(■- ^' + ( '&;r?::::::::r"' ) — •••)} ■ (35) In the simple case F = 1 + p£&& + ft 2^+|#+?L^4 (36) p-y Pi-Pi+P27y+p-2 the differential equation is a(3-Y + /y((a + (3 + 1 j 1 )x"^ - y}D- (1, F +p x 2h{ x ' h - 1 }^ 2 D (a, F = aflla + „ i <j±M±vy, + p i2 ^±a- a+ Pi + ^-/3+^^+j>i+p 2a!P ,-to + . . .) H \ y i\y Pilh+lh-yy+lh _(1 + p a ±M±Plr>', + « 2 « +ffi<* +ffi + JVff +i j i/? +Pi +V ?, + . . . .) I . (37) Puttino- x = 1 the expression on the right vanishes identically, therefore when [D (2) F] a:=1 is finite or convergent we have «/3[FU + pM + P-y+Pu which is dx w _ x=\ = . . . (38) i + « s '-'+j ) i/ i '^j ) i + = Pi(y- a -P-Pi) {1 +p a ±PiP + Pi PiPi+P2-yy+p2 y ' x iYy + i>2 a/3 ] Pi7 + p 2« + ?V a +Pi +P 3 -P +frP +Pi +P 2 + , .,(, . (39) 1 Pi-Pi+Pi-y+Pa-y+Pi+P* ') subject to the convergence of the series. When the progression of elements Pip% is 1 3 5 7 ]?<-, a y+ i, ,■) = 1 4 -%, + t ?; +1 /f + V + • ■ ■ • l-'-y+l l z -2---y + 1-y + 4 F(„ft !,,, , 1 + tf, ♦ '- + I ;f±ix. + -±i^»±M±lV + (40) In connection with this series we can obtain from (39) the identity 1 + {uf + (ST** + (sir - *"" s/l -l. »-l -g-l_,. »-l 2 -»-2 3 x-l 2 -a;-2 2 , » - V-n - 2H - 3 2 a>- l 2 a- 2 2 x- 3 2 ^ ) - (j: + n; -j + -^- -^ + -^-^ -^ + 122 2. 3 2 2 2 3 2 4 2 ' ' J { ' putting n = — x = m 2 , //t 4 m 4 -m 4 - 1 4 _ ^ a\y (2i)* " In the case when Pjjt^ form a geometrical progression and F([a] [/3] [<y]Xa;) denotes the series 1 + M [/3] A ,,n . M [« + *] [0] [0 + Qxym + ( ± 2 ) WW HPiHrKy+q ( } u p- 1 the following relations can be obtained : FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 37 F([a][fl[ r >r— "-<) = fr " ^L^^l^" ^ 1} f (M 08] fr - *1j»— '-') • (43). F([a]L8][y>v— ^) = ^Jr_^^L ] . F([a][ ^ ][y]py -a- 3 - F([a][/i][7>v — *) = * F([-«]C8][ 7 -a]py-/i) and other similar relations found by interchanging a and /3. The expression of F([u] [/3] [7]p y_<1 ^) and F([a] [/3] [y]p y ' a ~ p ~ l ) in the form of infinite products analogous to ^-\Zf™ is effected at once by the above relations. The investigation of these I have given elsewhere, but note the results here as of interest in connection with the Fundamental Hypergeometric Equation discussed in this paper. Particular cases of these series are r<[,-]) = (jf My* m 2 my* . i ! T([x + $]) \[x] [l]![as][a:+l] [2]! [x] [x + 1] [x + 2] l> 1]' if + [JlV 1 , Mf]'^ , l J (44 v i m rii!Hk+n r2iiTiira!+-nraj+2i ■ ' • ) " v r Bessel's Series : — Consider now a progression P-6 P-5 P-i P-A i'-2 V-l Po Pi Pi lh denote P1+P2+ ■ ■ ■ +P„ by (ra) and P0+P-1 + • • • • +P-,,+, by (-71) lhen the operator Apjjjja^D'"-'' + \{(1) -(»)-( -»)b 1 a ,1 'D (1 ' + X(n)( - ra)D u > = <£ . . (45> operating on A^"" gives A[(m 1 )((m 1 ) - (1)) + {(1) - (n) - ( - n)}(v h ) + («)( - ijflA^ which is So that if we operate on a series y = ^ Ax (m) $"J =-- 2t(™) - (»)] [(»») - ( " ?i >] Aa:( "" choose (//« T ) = (?j) and (m. 2 ) = (ra + 2) K) = ( M + 4 ) AA, +1 [(m, +1 ) - («)] [(m r+1 ) - ( - »)] = A' TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 2). 38 FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS, then ( ~(»+2) I J \ + [ (w + 2 ) -(»)][(n + 2) -(-»)] f and ( r (»+4) ) *» = iA { ^ + V +2) - W H(» + 2)- ( - »)] + ' ■ ' } ' ■ ■ (46) when the elements are Pi = Pi = P 3 = 1 Fo = P-\ = -P-2 = - l The series is Bessel's Series and <P'y = ^x 2 y is Bessel's Equation. In the particular case of the progression of elements PiP 2 .... being P-3 P-2 P-\ PO Pi Pi PS • ■ • • -7-5-3-1 1 3 5 . . . . andA= -1 I ~(n+2)s ,j.(«+4) 2 ' ) V = \ X (n + 2)*-n* {(w + 2)*-ra 4 }{(7i + 4) 4 -n 4 } " J " ( ' ( 39 ) III. — Magnetization and Resistance of Nickel Wire at High Temperatures, By Professor 0. GK Knott, D.Sc. (Read May 4, 1903. Given in for publication November 12, 1903. Issued separately March. 3, 1904.) In a recent paper published in these Transactions # I gave an account of certain experiments upon the change of electric resistance of nickel due to magnetization at different temperatures up to 100° C In the closing sentence of that paper I pointed out the advisability of trying to push the temperature up to 400° C, the temperature at which nickel loses its pronounced magnetic properties. This has been accomplished in the experiments now to be described. These refer, meanwhile, to the effect of longitudinal magnetization on the resistance of the wire. The experiments on the effect of transverse magnetization are still incomplete, and are reserved for a future communication. 1. The Apparatus used. — Since the temperature was to be raised to about 400° C, it was necessary to use asbestos for insulation. Accordingly, two exactly similar anchor- ring coils with nickel- wire cores were constructed. These coils were about 18*3 cm. in diameter. Eound the flat circular coil of nickel which formed the nucleus of the anchor-ring two independent layers of copper wire, carefully insulated throughout, were coiled, with the same number of turns in each. The ends of the two copper- wire coils could be joined in different ways, so that it was possible to have a strong current pass- ing through both, and yet to have, at will, either strong magnetization within or none at all. The magnetizing force could thus be removed at a moment's notice by simply reversing the current through one of the coils of copper wire, while the heating effect of the current on the whole coil could be maintained unaltered. To preserve as con- stant a temperature as possible during any one set of observations was of the highest importance, for the change of resistance due to a very slight change of temperature was sufficient to mask completely the change due to magnetization. This change was measured by means of a Wheatstone Bridge arrangement. The galvanometer was made of a convenient sensitiveness for the purpose ; and only when a very steady tempera- ture was obtained during a set of observations was the galvanometer in a steady enough state of approximate balance to render measurements possible. It was for the purpose of further reducing the disturbances due to changing temperature that two anchor-ring coils were used, with equal lengths of nickel wire as the cores, and with the same magnetizing or merely heating current flowing through both double coils of copper wire. * Vol. xl. pp. 535-545. TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 3). 7 40 PROFESSOR C. G. KNOTT ON MAGNETIZATION The nickel wires in these coils will be called L and M for ease of reference ; and the same letters may, if occasion arise, be used each to designate the corresponding anchor-ring coil as a whole. The wires L and M, with their stout nickel-bar terminals, formed two of the con- tiguous branches of the Wheatstone Bridge, the other branches A. and m being also made of nickel wire, so as to minimise the possibility of thermoelectric effects in the circuit. The resistances in B. A. ohms of the various branches at 13° C. were as follows, L and M being the resistances of the parts of L and M included in the magnetizing coils. Wire. L M X. /* Lo M Resistance . 31237 3-041 3-2413 3-1556 2-957 2-925 The lengths and number of turns of the four magnetizing copper coils were as follows : — Length of copper wire Number of turns L-coils Inner. Outer. 899 cm. 172 1151 cm. 172 M-coils Inner. Outer. 992 cm. 171 1237 cm. 171 In the experiments to be described it was the nickel wire L which was studied ; hence, to reduce the magnetizing current in amperes in the coil L to fields in magnetic C.G.S. units we must multiply by 7*52. The magnetizing current was measured on a Kelvin graded galvanometer, which was calibrated in situ by comparison with a Kelvin ampere balance. The two anchor-ring coils L and M were enclosed in a porcelain vessel, through the side of which the various terminals were led. A quantity of asbestos wool was packed round the coils so as to reduce as far as possible the convection currents of hot air when the vessel was heated. The heating was effected by means of one, two, or more Bunsen burners, according to the temperature aimed at. After several hours' heating the temperature of both coils became fairly steady ; and when the galvanometer indicated that a sufficiently steady temperature was reached, the necessary observations began to be made. The temperature of the wire L at any instant was indicated with great accuracy by the resistance of the wire, an independent experiment upon a wire cut AND RESISTANCE OF NICKEL WIRE AT HIGH TEMPERATURES. 41 from the same piece giving data by means of which the measured resistance could be reduced to centigrade degrees. It is known # that the resistance of nickel changes rather curiously with temperature, of which more hereafter. In the present case the resistance temperature graph is, roughly speaking, a long sloping S form of curve, and can be represented for interpolation purposes very approximately by three straight lines. Thus, to reduce the measured resistance r of the wire (3 ohms at 15°) to temperatures the formulse are t = 567r - 163-3 from 15° to 200° t = 33-9/' - 22-7 ,. 200° „ 350° t = 95-8r - 692-5 „ 350° „ 400° 2. The Method of Experiment. — The resistance changes due to magnetization were measured by deflections on a delicate galvanometer after the bridge was approxi- mately balanced, a steady current being supplied from a secondary cell in the battery branch. The galvanometer was calibrated in the following simple way. For any particular condition the current i through the galvanometer is given by the formula ~Di = (LfjL - MA)e ..... (1) where e is the electromotive force in the circuit, and D the well-known expression involving the resistances of the six conductors making up the Wheatstone Bridge. Its value is D = BG(L + M + A + fi) + B(L + A)(M + fi) + G.(L + M )(A + /*) + LM(A + /jl) + A/*(L + M) , where B is the resistance in the battery branch and G that in the galvanometer branch. In the present experiments B was 6*2 ohms and G was 1 73 ohms. The calibration of the galvanometer in each experiment was effected by making a known small change in A ; and the magnetic effect on resistance gave a slight change in L. We must therefore consider how the quantity D changes with slight changes of A and L. Differentiation of D gives d ® = BG + B(M + /x) + (G + /*)(L + M) + LM ^ = BG + B(M + /x) + (G + M)(A + /x) + V- The particular values of these quantities will vary from experiment to experiment according to the temperature of the branches L and M. For temperatures higher than the temperature of the air, L was always at a higher temperature than M, since it lay lower down in the porcelain vessel. To obtain the approximate balance on the Wheat- stone Bridge m was appropriately varied so that the quantity (L/x - MA) rarely differed more than l/400th from the value of Lm or of MA. When the difference was l/200th the spot of light was driven off the galvanometer scale. In the following table the values * See my paper on the Electrical Resistance of Nickel at High Temperatures, Trans. R.S.E., 1886 ; also a paper by W. Kohlrausch, Wiedemann's Annalen, vol. xxxiii., 1888. 42 PROFESSOR C. G. KNOTT ON MAGNETIZATION of the various resistances and of the quantities referred to above are given when the temperatures of L are about 15°, 180°, 300°. Resistances ■when the Temperatures Branches. of L are 15° 180° 300° 1) 6 2 6-2 6-2 G 1-73 1-73 1-73 L 3-12 6 10 M 3-04 5 8 A 3-24 324 3-24 H- 3 16 2-7 2-6 D 571-2 1009-9 2058-6 dD/dk 88-8 126-4 2344 dD/dL 89-9 96-4 141-7 Iii the calibration experiments the slight change in the resistance of X was always effected by putting in a resistance of 30 ohms in multiple arc with a small part of X whose resistance was - 5125. That is to say, the change d\ was a decrease of 0'008603 ohms, and d\/\= —0-002655. Returning now to equation (l), namely, D* = (V-MA)e ..... (1) we get, in consequence of the small change d\, the equation Ddi + idB = - UedX ..... (2) Similarly, when by application of the magnetizing field L is changed to L + £L, we find mi + i8D = + fieSL ..... (3) Substituting for i its value as given in (1) we get for (2) and (3) / La - MX dD\ -1 m.---«bl(m + -V- dx)\ ns . sm / L/a-MA d,D\ y (4) If, at the beginning of the experiments corresponding to equations (2) and (3), the bridge were accurately balanced, the current i would vanish and we should have or di 87 _ M d\ _ L dX A 8L ZL Si d\ L di X (5) This was the case in many instances, and in many others the value of i was so small as to make (Lm - MX) less than the thousandth part of L/m or MX. Under these AND RESISTANCE OF NICKEL WIRE AT HIGH TEMPERATURES. 43 conditions, taking the largest value of the ratios dD/Dd\ and dD/Ddh, namely, 89/572, we find D* = - ,/a(m + MX 5T |_) D8i= +^-^5t|oo), showing at once that the second terms in the brackets are negligible, and that equation (5) still holds. Finally, consider the most extreme case of all, in which the applied field was so great as to produce a change dh, which made the spot of light travel from end to end of the scale. It was necessary in this case to disturb the balance on the bridge, so that the initial current i produced a deflection 3 '5 times that due to the imposed change d\. That is to say, since the change d\ meant an alteration of 0*002655 in the value of L/« or MX in the experiment corresponding to equation (2), the initial value of i in this extreme case of the experiment corresponding to (3) was such as to make L/x-MX = 3-5 x -002655 L/x = -0093 L M and MX = 9907 L^ . Thus equations (4) become Bdi = - eMdk I T>U = + e/*8l/l - -0093 ^ ^) | This extreme case occurred only at the ordinary temperature of 15° C. Putting in the corresponding values of L, D, and dJ)/dJj, and taking the ratio, we find di __M.dk 1 Ji ~ J SL 1 - -0093 x 3-12 x 0-158 = _ ^ ** 1 4 x -9907 = - h '/ T A x -995 . X SL 1 - -00458 k SL Hence equation (5) is in error by 0*5 per cent. ; and it will be noticed that this is due mainly to the factor by which we pass from the ratio M/V to L/X. The effect of the second terms in the brackets of equations (4) is in all cases negligible. Hence in every case we may write di _ _ M dk Si /j, SL and in the great majority of cases use the more convenient form (5). • 3. Eeduction of the Observations. — The results embodied in Table A were plotted on a large scale, the change of resistance dh being plotted in terms of the field. In any one series of experiments the temperature varied a little throughout ; but it was -easy to apply slight corrections by graphical interpolation so as to obtain a series of isothermal curves. From these curves the values of cfiL were read off for the fields 2, 4, 6, etc., up to 34, and were then divided by the appropriate value L of the re- sistance of the nickel wire included in the coil. This was assumed to have the same 44 PROFESSOR C. G. KNOTT ON MAONKTIZATTON Tablk A. Containing the reduced results immediately deducible from the individual observa- tions of the experiments described above, and arranged in order of date, H being the magnetic force in the heart of the anchor-ring coil, t the temperature of the nickel coil at the instant, as determined by the measured resistance of the tvire L, and dL being the change of resistance due to the magnetization. H t dL H t dL H t dL (Feb. 13 ) (Feb. 16 > (Feb. 18) 33-6 14 0-0298 32-2 179-1 0-0262 32-6 127-5 0-0291 26-2 15-3 222 25-8 183-1 207 25-8 130 236 21-8 16-3 176 21-2 185 183 21-2 128-1 198 17-8 175 142 12-5 180-8 110 16-1 124 152 13-5 18-7 96 8-9 179 79 10-2 121-5 88 9-4 19-2 48 6-8 176-9 51 7-8 117 54 71 18-9 25 5 176-3 25 5-1 113-2 20 5-2 169 11 3-2 176 6 3-2 111-1 5 3-5 16-3 24 (Feb. 19) (Feb. 21) (Feb. 23) 30-8 2415 0-0173 30-4 317 0-0044 34 16-7 0-0296 24-8 242-7 150 24-4 322 20 26-6 197 238 20-7 242-1 133 20-2 322 25 21-8 20-4 194 15-3 241 107 14-9 319-7 21 16-6 18-7 126 12-3 240-5 88 11-9 322 19 13-0 18-1 96 10-1 239 5 76 10 323 14 10-8 17 68 7'1 237-7 54 7-4 323-5 11 8 16-3 31 4-9 236-1 27 4-8 323-5 5 5-8 15-3 14 3-6 234-8 16 4-2 14-6 6 (Mar. 12; (Apr. 1) (May 6) 31-8 279 0-0117 31-8 328 0-0043 31-8 299-3 0-0097 214 278-3 96 21-6 328 28 31-6 303-4 71 12-9 276-3 69 10-6 326-4 6 15-7 302-7 49 8-9 274-6 51 (May 10) 10-3 301 38 5-9 272-7 34 30-4 342-3 46 7-1 300 3 3-6 271 23 22-0 344-3 15 4-5 299 2 2-3 269-1 6 4-8 342-2 7 2-7 299 08 (May 14) • (May 15) (May 18) 32 17-9 0-0331 33-3 67-7 0-0301 34-1 16-3 0-0280 17-8 16-6 130 26-2 68-6 241 26-8 18-7 225 14-1 15-4 109 21-5 68-8 195 22 18 186 13 15-3 93 16-5 67-2 152 16-3 14-7 127 12-8 16-3 91 13 66 111 12-8 15-4 90 10-6 15-8 67 10-7 64-9 77 10-8 15 66 10 15-5 54 8-4 63-5 53 8-1 14-7 33 8 IB'] 34 6-1 61 26 4-7 14-4 7 65 14 1 19 4-1 62-3 11 4-4 13-8 5 3-4 10-7 21-6 60-3 60 63-5 6 81 206 AND RESISTANCE OF NICKEL WIRE AT HIGH TEMPERATURES. 45 ratio to the whole resistance L at all temperatures. It is possible, however, that with the strongest currents, whose heating effect is quite apparent in the slight rise and then fall of temperature during one set 01 readings as shown in Table A, the part of the nickel wire included within the magnetizing coil might be slightly higher in tempera- ture than the short parts outside the coil which joined the nickel wire to the stout nickel terminals. This would make the ratio L /L a little greater in the highest fields, so that the quantity c?L/L would be a little smaller. It is clear, however, that any small error due to this cause will not materially affect the broad conclusions to be drawn from the experiments. In Table B are given the final reductions, each column corresponding to a particular temperature, and each horizontal row to a particular field. The numbers entered are the increments of resistance per 1 00,000 ohms. Table B. Shoiving resistance changes per 100,000 ohms of nickel tvire in various longitudinal fields and at various temperatures. Magnetic Resistance Changes at Temperatures Field. 15° 65° 125° 180° 240° 280° 300° 328° 342° 34 1040 SI 6 621 475 253 141 86 45 5 32 970 770 594 450 244 138 83 44 30 901 729 565 -126 235 133 81 43 28 825 679 529 401 226 128 79 41 4 26 749 629 498 375 216 123 76 38 24 682 579 463 350 205 119 73 35 22 613 529 425 324 19-', 113 69 32 2 20 543 479 392 298 179 100 66 29 18 474 429 356 270 166 100 61 24 16 411 379 3 1 7 242 151 93 57 18 1-6 14 341 318 275 215 135 84 52 12 12 268 255 231 186 112 75 48 6 10 192 187 181 158 103 64 42 4 1 8 119 124 123 121 85 52 37 3 6 53 63 65 70 56 38 30 2 4 17 23 21 23 27 27 19 1 2 3 5 5 5 5-5 5 4 ... The numbers in the last column under temperature 342° C. were just measurable ; anything under 3 is, in fact, barely outside the errors of observation. 4. Discussion of the results. — These numbers give two sets of graphs — namely, the isothermals showing the relation between magnetizing force and resistance change at the various temperatures, and the isodynamics showing the relation between the 46 PROFESSOR C. G. KNOTT ON MAGNETIZATION resistance change and the temperatures in the various fields. These sets of curves, marked b, a, are given in the accompanying Plate. The first obvious result is the diminution of the resistance change in the higher fields as the temperature rises. Thus the effect in various fields at temperature 15° is from 200 to 300 times the effect at temperature 342°. So rapid is the final drop above 300° C. that we may safely regard the effect as practically non-existent at temperature 350° C. It is just at this temperature that nickel loses its strong magnetic properties, RESISTANCE CHANCES ACCOMPANYING MAGNETIZATION OF NICKEL AT HIGH TEMPERATURES .005 Temperature 200° the permeability being practically unity. Thus we learn that the change of resistance of nickel wire due to the application of a longitudinal magnetic field is mainly a function of the magnetization or induction in the material, and not of the magnetizing force. In fields below 5, there is first increase of the resistance change as the temperature rises. In fact, all the isothermals from 65° to 300° begin above the isothermal of 15°, and then cross it as the field increases. This is particularly well marked in the case of the isothermals 65°, 125°, and 180°. This phenomenon may be connected with the fact that, up to a certain limit, the induction curve for nickel rises more abruptly and reaches its ' wendepunkt ' in lower fields the higher the temperature. In other AND RESISTANCE OF NICKEL WIRE AT HIOH TEMPERATURES. 47 words, the first effect of rise of temperature is to increase the permeability in lower fields, probably because of the greater ease with which the molecular groupings assume new configurations. But anything which tends to increase permeability must tend to increase the effect on resistance. As the magnetization approaches its satura- tion value, rise of temperature diminishes the permeability, and rapidly so as the critical temperature of 350° is approached ; and very similar is the effect of rise of temperature on the change of resistance due to a given field. The isodynamic curves indicate the existence of a further peculiarity which declares itself at or near the temperature of 180° by a kind of cusp-like peak in the graphs of the higher fields. This peculiarity is also well brought out by calculating the differences between the resistance changes corresponding to the successive temperatures in the pre- ceding table, and dividing these by the change of temperature. These average differ- ences per degree will correspond to the mean of the extreme temperatures ; and their values for five of the fields are given in the following subsidiary Table C. Table C. Showing differences per degree calculated: from Table B. Magnetic Differences per Degree at Temperatures Field. 40° 95° 152°-5 210° 260° 290° 314° 335° 34 4-5 3-3 2-7 3-7 2-8 2-7 1-5 2-9 28 29 2-5 2-3 2-9 2-5 2-5 1-4 2-6 20 1-3 1-5 1-7 2-0 1-8 2-0 13 1-4 10 o-oi o-oi 04 0-9 1-0 IT 1:4 0-2 6 - 0-02 - 0-003 -0-9 0-7 0-45 0-4 TO From these few examples we see that there is at or near the temperature 200° a peculiarity which shows itself by an increase in the rate per unit rise of temperature at which the resistance change due to a given field is diminishing. 5. Comparison with results formerly obtained. — When we compare the results here given with those obtained in the earlier experiments a considerable discrepancy declares itself. From the results given in the final table in the earlier paper (Trans. R.S.E., vol. xl. p. 543), we readily find by interpolation the resistance changes at the three temperatures 127, 57"5, and 93'5, corresponding to the fields 30, 22, and 14. Then, from the table given above (p. 45) we can interpolate values corresponding to the same fields and temperatures. These are compared in the following short table (D), the earlier and later results being distinguished by the Roman numerals I. and II. TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 3). 48 PROFESSOR C. G. KNOTT ON MAGNETIZATION Table D. Field. 30 12° to 15° I. II. 57 I. 5 "5 II. 93 I. '•5 II. 349 901 376 754 405 647 22 215 613 233 543 260 477 14 109 341 121 321 140 297 When this discrepancy was noticed, the first idea was that some fundamental mis- take had been made in reducing one of the series of experiments. The mistake might have been made either in calculating the field in C.G-.S. units or in calculating the changes of resistance.* It is important to make sure that no such mistake has been made. I shall therefore reproduce exactly the observed numbers in two distinct experiments, one made on 19th June 1902 with the first apparatus, the other made on 13th February 1903 with the second apparatus. Anchor-ring coil I. had 846 turns, with a mean radius of 3 "6 cm. ; and anchor-ring II. had 344 turns, with a mean radius of 9*1 cm. Hence the reducing factors which enable us to calculate field from current are in the ratio of about 6*2 to 1. Now, in the two experiments mentioned above, the measured deflections of the currents on the Kelvin graded galvanometer were, respectively, 9*9 with the magnet at mark 44'35, and 22 with the magnet at mark 16 ; that is, the currents were in the ratio of 44*35 x 22 to 9'9 x 16 ; that is, about 6'3 to 1. Hence in these two experiments the fields were nearly the same, somewhere in the neighbourhood of 26. The following are the readings as jotted down in the experi- mental note-book, with the deflections and differences immediately deduced from them. Experiment I. — Field = 26*5. ( 1 ) Calibration of galvanometer. Shunt in \. inf. 20 inf. 20 inf. Reading on Scale when Current is Direct. Reversed. Direct. 134 166 133-3 165 132-7 173 141-2 174 142 174-5 134 165-7 133-1 165 132-6 Deflection. + 39 -24-7 + 40-8 -23 + 41-8 Difference of Deflections. 64-8 64-3 64-55 * The constancy of the resistances of the various coils in use and the steadiness of the results obtained in all cases quite preclude the possibility of any error in the estimation of the constants arising from faulty insulation. AND RESISTANCE OF NICKEL WIRE AT HIGH TEMPERATURES. (2) Measurement of resistance change due to applied field. 49 Direction of Field. Deflection when Field is Off. On. Off. Difference. + 113 169-8 114-4 56-1 - 117 173 119-3 55-4 + 121 176 120-9 55 122 177-9 122-5 55-5 mean = 55*5 Experiment II. — Field = 26*2. (l) Calibration of galvanometer. Shunt in A. Reading 01 Direct. 1 Scale when Current is Reversed. Direct. Deflection. Difference of Deflections. inf. 134 162-7 134 -28-7 30 169 127-8 168-5 + 40-95 69-96 inf. 133-5 163 133-7 -29-4 30 169 127-7 169 + 41-3 70-53 inf. 133-9 162-9 1338 - 29-05 mean = 70-25 Magnetic Field. off on- off on + off (2) Measurement of resistance change. Reading on Scale when Current is Direct. Reversed. Direct. Deflection. 150-5 239-8 141-5 231-5 136-8 146-5 57-5 155-9 64-5 160 149 234-7 140 229 136 + 3 + 179 - 15 + 165 - 23 •75 •75 •25 •75 •6 Difference of Deflections. 185-5 186-2 mean = 185 '9 In the calibration experiments conducted as described on p. 41, the column headed ' shunt ' indicates that in the second and fourth lines the resistance X was slightly altered in value by joining in multiple arc a fairly large resistance and a small part of A. In Experiment I., 20 ohms resistance was joined up with a resistance of 0"3085 ; in Experiment II., resistance 30 was joined up with resistance - 5125. The 50 PROFESSOR C. G. KNOTT ON MAGNETIZATION corresponding changes in A* were 0-0909/20-3015 and 0-2627/30-5125. Thus we find by equation (4) : — in L in II. rlL/L = 0909 55-5 x 2 20-3015x3-089 ' 64"45 2627 185-9 = -0025 ; 30-5125 x 3-2413 70"25 = -00703 Notice that in I. (1), II. (l) and (2), the deflections are really double the true values, whereas in I. (2) the deflection is given at once. For in the last case the approximate balance is altered by the magnetizing force being put on. In the other cases the deflections are due to the reversal of the current supplied by the single cell in the battery branch of the Wheatstone Bridge. The two cases here given in detail prove that there can be no doubt as to a difference of effect under apparently similar magnetic conditions. The nickel wires used in the experiments were cut originally from the same piece of wire. The only difference between the two forms of apparatus lay in the manner of winding. In the first small anchor-ring the nickel core was a small compact closely-wound coil of twenty windings of silk-covered wire ; in the second large anchor-ring the nickel core was a loosely-wound coil of some 10 or 11 turns, with asbestos paper interwoven. It is possible that in the compactly-wound coil the inner turns were screened from the full magnetic action of the applied field by the outer windings. This view receives some corrobora- tion from the manner in which the discrepancy established by the figures given above diminishes as the temperature rises. Taking the ratios of the corresponding changes in II. and 1. we get the following results : — Ratio of Resistance Changes (II : I) at Field. 12°-15° 57°-5 93°-5 30 2-58 2-01 1-60 22 2-85 2-33 1-83 14 313 2-65 2-12 Thus the measured effect in the earlier experiment deviates more from the same effect in the later experiment the lower the field and the lower the temperature. But this is j ust what would be expected if the discrepancy were due to magnetic screening, which is well known to become less evident in higher fields. There are no experiments, so far as 1 am aware, as to the effect of temperature on the screening effect ; but we have every reason to expect that it will diminish as the temperature rises. The In the earlier paper X was called n ; its value was less than the value of A. in the present paper. AND RESISTANCE OF NICKEL WIRE AT HIGH TEMPERATURES. 51 question here raised would probably repay further experimental investigation ; and it obviously suggests a new method for studying magnetic screening. 6. Comparison with other peculiarities of nickel at 200° C— That some kind of peculiarity should occur at about this temperature was not unexpected. It was indeed with the expectation of getting some such effect that these experiments were originally planned more than a dozen years. It was my good fortune as an undergraduate to assist the late Professor Tait in the thermoelectric investiga- tions which occupied his attention during the years 1872-4.* Probably the most remarkable results established by these investigations were those in connection with iron and nickel. The thermoelectric lines for all metals save iron and nickel are approximately straight through great ranges of temperature. Their inclinations in the properly constructed thermoelectric diagram give the Thomson Effects in the corre- sponding metals. In every case of a pure metal except those mentioned, the Thomson Effect retains the same sign throughout. In the case of iron and nickel, however, it changes sign — at a dull red heat in the case of iron, and at about 180°-200° in the case of nickel. But the nature of the phenomenon is the same in both. The Thomson Effect, which is negative at ordinary temperatures, becomes positive at higher temperatures ; and finally, when the temperature is raised still higher, negative again. The second change of sign occurs in each case at the temperature at which the metal ceases to be strongly paramagnetic. In the case of iron, another phenomenon is known to occur at the temperature of dull red, namely, the sudden expansion during cooling discovered by Gore, and the accompanying reglow discovered by Barrett. No similar effect has been observed in the case of nickel, possibly not because it does not exist, but because the temperature is too low to admit of a visible ' reglow.' In any case these phenomena point to a curious molecular change occurring both in iron and in nickel at a temperature well below that at which the magnetic permeability becomes unity. Mention has already been made as to the rather curious manner in which the resistance of nickel changes with temperature. In my paper on the electric resistance of nickel at high temperatures, referred to above, this peculiarity is established, and the difficulty in working at high enough temperatures prevented me establishing the existence of the same peculiarity in iron, although there was indication of its existence. This, however, was done shortly afterwards by W. Kohlrausch. The peculiarity in the case of nickel is shown by the interpolation formulae given above, p. 41. The rate of increase of resistance with temperature undergoes a sudden increase at a temperature of about 180°-200° C, and then diminishes as abruptly again at about 400° C. Once again, then, we have another set of phenomena indicating a peculiar molecular change in nickel at 200° as well as at 400°. In the present investigation the relation that is being studied involves the measure- * See Tait, Trans. Roy. Soc. Edin., vol. xxvii. pp. 125-140; also Scientific Papers, vol. i. pp. 218-233. TRANS. ROY. SOC. EDIN., VOL. XLL PART I. (NO. 3). 9 52 PROF. KNOTT ON MAGNETIZATION AND RESISTANCE OF NICKEL WIRE. merit of small changes, which are indeed changes of the second order, namely, the change per unit rise of temperature of the change due to a given applied field. It would be utterly impossible, in the present state of knowledge regarding molecular groups, to make any prediction as to how the molecular change indicated by the thermoelectric and resistance peculiarities should show itself in the present case. A glance, however, at the isodynamics, with their cusp-like jpoint in the higher fields, and a tendency to a maximum in the lower fields, seems to indicate some peculiarity at this critical temperature of 200°. What seems to be indicated is, that about this temperature the change of resistance with magnetization begins to fall off more quickly as the temperature is raised. ( 53 ) IV. — The Glacial Deposits of Northern Pembrokeshire. By T. J. Jehu, M.D. (Edin.), M.A. (Camb.), F.G.S., Lecturer in Geology at the University of St Andrews. Communicated by Professor James Geikie, LL.D., F.P.S. (With a Plate.) (Read February 15, 1904. Issued separately April 9, 190:1.) CONTENTS. PAGE I. Introduction 53 II. Previous Literature 54 III. Physical Features and Geology of the District 56 IV. Description of the Deposits 1. Tlie Lower Boulder -Clay 2. The Sands and Gravels 3. The Upper Boulder-Clay and Rubbly- Drift ... . . V. The Boulders and Erratics VI. General Conclusions page 63 63 68 74 77 82. I. Introduction. The area embraced in this paper consists of that part of Pembrokeshire which lies to the north and north-east of St Bride's Bay. Bounded on the west by St George's Channel and on the north by Cardigan Bay, it extends to the north-east as far as the mouth of the river Teifi, near Cardigan. That part of the country which lies in the immediate neighbourhood of St David's has, through the laborious researches of the late Dr Hicks and others, become well- known to geologists, and may now be regarded as classic ground. The solid geology of this promontory has given rise to much discussion, and has, perhaps, attracted more attention than that of any other part of the Principality. The reason for this great interest is to be sought in the facts that the rocks of this area are of a very great antiquity, and that the sedimentary series contain the remains of some of the earliest organic forms yet found in the earth's crust, whilst the igneous rocks are also displayed in great abundance and variety, and present us, in the words of Sir Archibald Geikie, with "the oldest well-preserved record of volcanic action in Britain." The geology of the district lying immediately to the north-east of the St David's promontory has not been the subject of so much attention, but the investigations carried on by De la Beche and the other officers of the Geological Survey before the middle of last century have recently to some extent been revised by Mr Cowper Eeed, and his results are published in a paper entitled "The Geology of the Country around Fishguard," which appeared in the Quart. Journ. Geol. Soc. (vol. li., 1895, p. 149). But while so much has been written concerning the ancient rocks of this country, very little attention has been paid to the more recent geological deposits. Owing to TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 4). 10 54 DR T. J. JEHU ON the facts that geographically the region lies further south, and that it does not present such a great elevation of the land above sea-level, it is not to be expected that Pembrokeshire will show such marked traces of the former presence of glacial conditions as are to be met with in North Wales. Nevertheless, it has been known for a long time that this region is to a large extent covered by more or less loose and uncon- solidated material, which is usually spoken of as Drift. And references are found scattered in the geological literature of the district relating to travelled boulders and other possible glacial phenomena there seen. But hitherto no attempt has been made to give a connected description of the glacial deposits with a view to the unfolding of the sequence of events which occurred during and after the Glacial epoch, and of correlating the results obtained by an examination of this area with those derived from a study of glacial deposits in North Wales and other regions. The need for further investigation will be evident to anyone who compares the map (plate i.) in Professor James Geikie's work on The Great Ice- Age, illustrating the British Isles during the Epoch of Maximum Glaciation, with the late Mr Carvill Lewis' " Sketch Map of England and Wales showing the Edge of Land Ice," which is reproduced in Professor Bonney's Ice- Work. In the former the southern boundary of the great ice-sheet is made to pass beyond Wales and run along the Bristol Channel ; and the northern ice which overwhelmed Anglesea is marked as crossing the western end of the Lleyn promontory of Carnarvonshire, and, joining the Irish Sea, it fills up St George's Channel and crosses the extreme tip of Pembrokeshire at St David's Head. In the latter the land-ice is shown as not extending over the whole of South Wales to the Bristol Channel, but with its southern edge extending no further south than is indicated by a line drawn eastwards from the St David's promontory, and the glaciation of Northern Pembrokeshire is attributed solely to local ice — the northern ice apparently extending no further south off the Welsh coast than the Lleyn promontory. The results obtained during the investigations carried on by the present writer will at any rate serve to settle the dispute with regard to the southward extension of the Northern or Irish Sea Glacier. II. Previous Literature. References to the surface deposits and surface features of Pembrokeshire are meagre and scanty in the extreme. Sir R. Murchison, in The Silurian System (p. 520), makes the following remarks : ' The detritus which appears on the surface of most parts of Pembrokeshire is of a simple character and, as in other parts of South Wales, is of local origin. It consists of fragments of greenstone, porphyry, carboniferous grits, etc., all of which can be traced to the various mountains forming the crest of the country. In some parts this detritus is exceedingly coarse. ... In other tracts, as north of Haverfordwest, we meet with finely comminuted gravel ; but this is rare." THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 55 Symonds, in his book entitled Records of the Rocks (p. 53), refers to the fact that the country near St David's must in former years have been largely covered by boulders. These have now to a large extent been cleared away from the surface and used for building rough walls which serve the purposes of fences and hedges. He adds that in his opinion " these boulders are all local, and have travelled over a slope of ice and snow which once reached from the Trap Hills of Precelly down to the sea." And in another place (p. 181) he speaks of the Precelly hills in North Pembrokeshire as being " studded with ice-carried boulders, which were used for cromlechs and tumuli by a prehistoric race of men." Messrs Howard and Small, in their " Geological Notes on Skomer Island," which appeared in the Trans. Cardiff Naturalists' Soc. (vol. xxviii., 1896), assert that distinct evidences of the action of ice are seen on the mainland opposite Skomer. And on Skomer Island itself blocks are found which have travelled from the St David's district and some possibly from North Wales. Flints were also seen scattered about. Professor M'Kenny Hughes, in his paper " On the Drifts of the Vale of Clwyd " (Quart. Journ. Geol. Soc, vol. xliii., 1887), remarks that "the low-lying plateau at St David's is covered by a gravel containing flints." But he found no traces of shells there. In a paper by Professor Bo^ney " On the So-called Diorite of Little Knott (Cumberland), with further Remarks on the Occurrence of Picrite in Wales" (Quart. Journ. Geol. Soc, vol. xli., 1885), some observations are found communicated by Dr Hicks relating to the glaciation of the St David's region. A boulder of picrite was found on the promontory just to the east of Porth-lisky, " resting immediately on Dimetian rock, surrounded by an uncultivated area overgrown by gorse and heather." The striae along the coast are said to run usually from north-west to south-east. He adds : " But it is clear that very many of the boulders scattered over it must have come from the high land in the north-east of Pembrokeshire, the Precelly range. There is ample evidence of local till, and in places (at considerable elevations) of marine sand with transported boulders, fragments of flint being common among them." Dr Hicks was of opinion that " this points to the derivation of some of the materials, including possibly certain boulders, from a north-west source." The most important communication which has appeared on this subject is a very short report read by Dr Hicks at the Cardiff meeting of the British Association in 1891, " On the Evidences of Glacial Action in Pembrokeshire, and the Direction of Ice-Flow." This report is reproduced in the Geol. Mag. for that year. He there refers to the presence of ice-scratched rocks and of northern erratics in the district. The direction of the glacial striae and the probable presence of erratics from North Wales and from Ireland " would tend to the conclusion that glaciers from these areas coalesced in St George's Channel, and that the ice which overspread Pembrokeshire was derived from both these sources, as well, probably, as from a flow extending down the Channel from more northern areas." By far the majority of the boulders are said to be of local origin, but he notes a large boulder of granite and another of picrite found on Porth- 56 DR T. J. JEHU ON lisky farm. " The picrite boulder has been shown by Professor Bonney to resemble masses of that rock exposed in Carnarvonshire and Anglesea, and the granite boulder, which before it was broken must have been over 7 feet in length and 3 to 4 feet in thickness, is identical with a porphyritic granite exposed in Anglesea, but not found anywhere in Pembrokeshire." He found clear evidences showing that St Bride's Bay was overspread by a great thickness of drift from the hills immediately to the north. " The intervening preglacial valleys were also filled by this drift, and the plains and rising grounds up to heights of between 300 and 400 feet still retain evidences of its former presence, and many perched blocks." Chalk flints were found at heights of over 300 feet, and have probably come from Ireland. He refers also to the crushing and bending of the strata at places, and to some well-marked examples of " crag and tail," but he does not locate these phenomena. The late Professor Prestwich, in his paper on " The Raised Beaches and ' Head ' or Rubble-Drift of the South of England" (Quart. Journ. Geol. Soc, vol. xlviii., 1892), refers to the possible occurrence of this rubble-drift on the coast of Pembrokeshire. He thought that he had detected traces of a raised beach and ' head ' near Porth Clais, and again at Whitesand Bay. Professor Bonney, in his Ice- Work (p. 161), states that " In Pembrokeshire and the adjoining districts erratics are often abundant, as may be seen near St David's. At present no systematic attempt has been made to trace them up to their sources, but they have probably come from the higher ground inland, that is to say, roughly, from the north-east." And again (p. 165) he refers to the possibility that the northern ice travelled down the bed of the Irish Sea, and perhaps ultimately overflowed St David's Head. Dr Wright, in his book on Man and the Glacial Period, remarks that "At St David's peninsula, Pembrokeshire, striae occur coming in from the north-west, and, taken with the discovery of boulders of northern rocks, may point to a southward extension of a great glacier produced by confluent sheets that choked the Irish Sea " (p. 143). Mr Cowper Reed, in the paper already referred to, mentions the fact that "drift or boulder-clay causes a difficulty in tracing the boundaries or determining the characters of the underlying beds " in the Fishguard district. III. Physical Features and Geology of the District. The county of Pembrokeshire lies in the extreme south-west corner of the Principality, and that part of it which is under consideration in this paper extends further to the westward than any other part of England and Wales, with the exception of the extremity of Cornwall. The promontory of St David's is washed on three sides by the sea which has eaten into the land so as to give rise to a variety of recesses and bays. It is the presence of hard igneous rocks that has enabled it to resist the ceaseless action THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 57 of the waves, which, owing to the direction of the prevalent winds, often beat upon the coast with great fury. The softer Lower Palaeozoic slates and Carboniferous shales to the south have succumbed to the encroaching sea, and there given rise to the broad and wide bay of St Bride's. The islands lying to the west consist partly of hard igneous rocks, and no doubt were once joined to the mainland. The coast scenery is magnificent, and throughout this region the rocky cliffs rise steeply out of the sea, and sandy beaches are only found here and there, such as at Whitesand Bay, Abereiddy Bay, Abermawr, Goodwick Bay, and Newport Bay. The cliffs in places are nearly perpendicular, and everywhere exhibit excellent sections of the rocks. The outline of the coast is very jagged, reefs and stacks of rock sticking out here and there, whilst, on the other hand, the sea has penetrated in so as to form caves and coves, and small narrow ria-like channels, such as that seen at Porth Clais at the mouth of the Alan river, and that at the mouth of the Solva river. A silted-up estuary occurs at Abermawr to the west of Strumble Head, and a larger one on the east side at Goodwick Bay ; they now form swampy ground. The only other estuaries of importance lie on the north-east side of Strumble Head, at the mouths of the rivers Gwaen, Nevern, and Teifi. That of Gwaen is also somewhat Wa-like in character. On the north coast Strumble Head is a prominent feature and stands out boldly to sea, and a little further to the north-east a small but well-marked headland occurs at Dinas. The one-inch Geological Survey maps of this part of Wales were prepared before the end of the first half of last century, and no revision has yet been made. Since that time a more complete knowledge of the fossil contents of the sedimentary series has been obtained, and improved methods for the study of igneous rocks, especially with regard to their microscopic structure, have been introduced. The need for a fresh survey is generally recognised, but much new light has been thrown on the geology of St David's promontory during the last thirty years through the researches carried on by Dr Hicks, Sir A. Geikie, and others. And Mr Cowper Reed has examined and described within recent years the geology of the area around Fishguard. But the region is a very complicated one, and much of the geology still remains obscure. The rocks of this part of Pembrokeshire are almost entirely of Lower Palaeozoic age, and a remarkable variety of both the igneous and sedimentary kinds is there displayed. In the St David's region a very full development of Cambrian rocks is exhibited, and these are underlain by a series of volcanic rocks — both series often showing signs of meta- morphic changes. The base of the Cambrian was taken by Dr Hicks to be marked by a conglomerate in which are enclosed pebbles of the underlying rocks. The volcanic tuffs and breccias which underlie the conglomerate were taken to be pre-Cambrian. Underneath these again comes a granitoid mass, which he regarded as still older. Later, the district was visited by Sir A. Geikie, who, after an examination of the ground, arrived at the conclusion that the granite is an intrusive mass, and that there is no break between the Lower Cambrian rocks and the volcanic series underlying them. 58 DR T. J. JEHU ON The granite covers an area lying immediately to the south of St David's, and there is another wedge-shaped mass a little to the south-west, reaching the coast on the eastern side of Porth-lisky. The St David's mass graduates into a spherulitic quartz- porphyry and felsite at its northern end. The granite is surrounded on all sides but the south by rocks of the volcanic series which are marked as Andesites on the index-map. These form a ridge running E.N.E. and W.S.W., stretching from Llanhowell, past the city of St David's, to reach the coast at the southern end of Ramsay Sound. Two detached masses are marked further east, about Llanreithan. The volcanic group consists largely of bedded tuffs ; but lavas also occur, and give rise to prominent crags to the west of St David's. Dykes and sheets of diabase traverse the other formations. These igneous rocks are flanked on the west and south-west by the Cambrian conglomerate, and this is followed by green, purple, and gray flaggy sandstones, with intercalated red shales. Towards the base, fragments of Olenellus were found by Dr Hicks. In a south-east direction these are followed by the gray and black flagstones and shales of the Menevian series, and these again by gray and bluish flagstones and slates of the Lingula Flag series. Still further eastwards, a small tract of Tremadoc beds is found near Tremainbir. Beds of the Menevian and Lingula Flag series also occur at the south of White- sand Bay. The Lingula flags run as a continuous band from the bay inland as far as Crug-las, and on the north a band of Tremadoc flagstones and earthy slates runs parallel to a point south of Abereiddy Bay. The north coast from St David's Head to Abereiddy is made up of slates and shales and flagstones of Arenig age. Masses of gabbro occur at St David's Head, and a little east of this, diabase masses, giving rise to rugged eminences, are seen. North-east of Abereiddy the Llandeilo flags succeed the Arenig series, and consist of black slates and flags, sometimes calcareous, and some felspathic tuffs. Numerous bands of " felspathic trap " are seen to occur in the tract bordering Abereiddy Bay. Eastwards from the St David's promontory, right into mid- Wales, the sedimentary rocks are marked in one colour on the Survey maps, and are referred to as " Lower Silurian (including Upper Silurian not yet separated)." They consist of shales, slates, and gritty sandstones, with some flagstones. Mr Cowper Reed found that beds of Llandeilo and Bala age form Strumble Head. Immediately around Fishguard we meet with Llandeilo and Arenig beds, and further east with Upper Llandeilo and Bala beds again. Dinas Island is composed of sandstones, slates, and conglomerates of Llandovery age. The country around Fishguard is rich in igneous rocks. Felsites, tuffs, and agglomerates contemporaneous with the Llandeilo and Bala beds occur, and intruded into these are sills and masses of " greenstone." The latter include basalts, dolerites, diabases, and gabbros. THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 59 Contemporaneous volcanic rocks and intrusive sheets of diabase occur in the district "between Fishguard and Newport, and further south in the Precelly mountains strips of felspathic rock are indicated on the Survey maps. East of Newport the igneous rocks die out. Blown sands are heaped up at places on the coast, and are generally full of land shells. To the east of Whitesand Bay they rise to heights of 150 feet and are pro- longed inland for nearly a mile, giving rise to the tract known as " The Burrows " or " Towyn." They are also seen at Abermawr, and cover an extensive area at Newport, and again at the mouth of the Teifi, where they form dunes known as the " Towyn Warrens." Towards the western end the country, as seen from a height, presents the aspect of a flattened tableland or plateau, having for the greater part of its extent no great elevation above sea-level, but here and there having rocky knobs and masses jutting out, especially towards the western and north-western extremities of the promontory. The view, as seen from one of the hills on Ramsay Island, suggested to Sir Andrew Ramsay that in this part of Wales we have the remnant of an old plain of marine denudation, which is continued into Cardiganshire and further inland. The rugged masses of Cam Llidi and Pen Berry, which rise so boldly just east of St David's Head, and have relatively the appearance of considerable mountains, only attain heights of 595 feet and 576 feet respectively. Further east the rough eminences which stand out so prominently on Strumble Head are mostly under 600 feet in height — Garn Gelli alone exceeding that limit and attaining 625 feet. Garn Fawr, famous for the remains of ancient fortifications there found, is just short of 600 feet high. Between Fishguard and Newport is a ridge of high land, reaching elevations of over 1000 feet at Mynydd Melyn, Mynydd Caregog, and Cam Ingle. To the south of these rise the Precelly mountains, which have a somewhat smooth outline and attain heights up to 1500 feet. From Newport to Cardigan the country is hilly, but not mountainous — the highest point being Pen Creigiau, which is 642 feet above sea-level. The highest ground is formed of diabase and other intrusive igneous masses ; the volcanic rocks occupy ground above the average level, while the low-lying ground consists of the more easily denuded slaty and shaly beds. Passing from west to east, the land generally becomes more hilly, and the higher grounds from Newport to Cardigan are composed of hard sandstones or arenaceous slates. The main streams of the district occupy pre-glacial valleys, and have cut their way through the drift which once filled them. The estuaries of the rivers Solva, Alan, and the Gwaen are Wa-like in appearance, and it is probable that an arm of the sea once extended for some way up the lower course of each ; for the estuaries are trench-like, with steep rocky walls on either side for a considerable distance inland, and it is hardly conceivable that these have been cut out altogether by the action of the streams. On the northern coast there are two peninsulas of a very peculiar character, for they are separated from the mainland by trench-like valleys, which, though now never 60 DK T. J. JEHU ON occupied by the sea, look as if they had been so in comparatively recent times. They are both spoken of locally as islands : one lies between Abereiddy and Porth Gain, and is known as Barry Island, and the other is Dinas Island, west of Fishguard. The valley between Dinas and the mainland is particularly striking, being only a few feet above sea-level, whilst that at Barry Island is not much less noteworthy, though its bottom attains a somewhat higher level. These peculiar valleys, together with the indications shown of the former presence of the sea up the inlets at the mouths of many of the streams, and the occurrence of swampy estuaries such as those seen at Abermawr and Goodwick, seem to point to a slight rise of the coast within recent times, causing a retreat of the sea. But at the present day the sea seems to be gaining once more on the land. At several places along the coast peat is seen at low tide, and in most of the bigger bays evidences of buried forests are sometimes seen. Giraldus Cambrensis, who wrote in the twelfth century, says with regard to St Bride's Bay at Newgall : — " When Henry II. was in Ireland an unusually violent storm on that sandy coast blowing back the sand discovered the appearance of the land concealed for so many ages ; stumps of trees standing in the sea, with the marks of the hatchet as if done but the day before, a very black earth and wood like ebony, so that it appeared not so much like a sea-coast as a grove."* And George Owen, in The Description of Pembrokeshire, written in 1603, gives an account of a very similar occurrence. He says : " About xij or xiij yeeres past were seene on the sandes at Newgall, by reason as it seemeth the violence of the sea or some extreeme freshe in the winter, washed awaye the sandes (w ch dayelye is and was overflowen with the tyde), soe lowe that there appeared in the sandes infinitte nomber of buttes and trees in the places where they had been growinge, and nowe euerye tyde overflowen : there appeared the verye strookes of the hatched at the fallinge of these tymber, the sandes being washed in the winter, the buttes remained to be seene all the sommer folio win ge, but the next yeere the same was covered againe with the sandes : by this it appeareth that the sea in that place hath intruded upon the lande." t Again about sixteen years ago a big storm washed away the sand and exposed roots of great trees in Whitesand Bay. Huge logs of oak trees were carried away by the neighbouring farmers, some of which are still stored, and were shown to the writer. Twigs and branches of hazel were found in abundance, although no hazel grows now near St David's. The writer is also informed that horns of deer were picked up- Similar evidence of a buried forest has been discovered in Goodwick Bay, and all along the coast up Cardigan Bay. All this reminds one of the old Welsh tradition regarding a great inundation of a land called Cantref Gwaelod, situated in the region now covered by Cardigan Bay, which is usually attributed to the fifth century. There is also a very old tradition that St Bride's Bay was formed by an inrush of the sea. * Rolls, ed. vi. 100. t Page 247 in the Cymmrodorion reissue. THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 61 Referring to some of the shallow valleys and peculiar bays seen in Pembrokeshire, Sir Andrew Ramsay, in his essay " On the Denudation of South Wales and the adjacent Counties of England," contributed to volume i. of the Memoirs of the Geological Survey t says (p. 329) : "In numerous instances valleys opening to the sea end in bays, and it will be found that not infrequently the headlands on either side of these bays were composed of comparatively unyielding materials. Depress the land, and these valleys become arms of the sea ; raise it, and the bays become a continuation of the valleys. If Pembrokeshire were elevated but 60 or 70 feet, Milford Haven would become a shallow valley of this nature, with occasional pools or small lakes in its hollows, through which would wind the water now flowing into the haven." And similarly, if the county was depressed but 60 or 70 feet, the valleys at the loAver part of their course would form sea-inlets like Milford Haven, and Barry and Dinas would be separated as islands from the mainland. It is practically certain that movements of elevation and of depression have taken place within comparatively recent times, but it is a difficult problem to ascertain the extent and duration of these movements. The land of the St David's promontory was long ago described by Giraldus Cambrensis as a " stony, barren, unimprovable territory, undecked with woods, undivided by rivers, unadorned with meadows, exposed only to winds and storms." Since his time the land has yielded somewhat to the continuous treatment of genera- tions of farmers, and a great part of it is now under cultivation. But much rough uncultivated moorland still remains, which in places is overgrown with gorse, and sometimes shows a boggy nature. Such are the commons seen round about St David's, and also further east. Parts of them are occupied by shallow sheets of water, as at Trefeithan common and Dowrog common. As a rule these commons are of a clayey nature, and in places some peat formation is seen. Peat becomes more evident in the country lying to the south of Fishguard and south-west of the Precelly hills, and it is dug even right up on the Precelly hills themselves. The country from Newport to Cardigan is well cultivated for most part. Throughout the area under consideration in this paper much of the soil is of a distinctly sandy nature. Hence the land is very dry and needs much rain. It is an old saying in this part of Pembrokeshire, that " in summer rain every day is too much, and every second day too little." The greater part of the land is covered by a blanket of superficial material, which may all be included under the name of Drift. This is somewhat variable in character, but near the surface a sandy element seems to predominate. However, as traced laterally, the sand often passes abruptly into clay or clayey loam, and vice versa. This drift near its upper part is usually stuck full of boulders and rock-fragments of all kinds, and of all shapes and sizes. Good sections are seen along the coast in some of the bays, but it is very rarely that one meets with a good exposure inland. Smoothed, polished, and ice-scratched boulders can be picked out of the drift in plenty, and occur throughout the district. It is more difficult to meet with examples of striated and TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 4). 11 62 DR T. J. JEHU ON smooth rock-surfaces, and this is partly owing to the fact that so much of the country is covered by the superficial accumulations ; and the igneous rocks which project above the surface are so weathered and worn from exposure for such a great length of time, that any marks of glaciation which they may once have shown must have become almost entirely obliterated. Reference has already been made to Dr Hicks' observations on the striae seen on the rock-surfaces at places along the coast near St David's. The best example noted by the present writer is at AVhitesand Bay, just at the northern corner. Here the rock, as it appears from under the cliff of drift, presents a distinctly hummocky appearance, and is smooth, polished, and well striated — the striae having a course from north-west to south- east. It is interesting to note that out of the drift above, striated stones can be readily picked : these are generally sub-angular, with blunted angles and somewhat smooth and rounded edges. In the small valley coming down near Porth Melgan, and separating the rocks of St David's Head from Cam Llidi, a pavement of sedimentary rocks has been exposed by the removal of some turf, and this pavement shows distinct marks of glaciation. Another very good example of a glaciated rock-surface is seen quite at the other end of the district at Gwbert, on the coast to the north of Cardigan. Here, emerging from beneath the drift again, a smooth and striated rock is seen — the direc- tion of the striae being from a little west of north to a little east of south, showing that the ice must have come on to the land from the region of Cardigan Bay. The igneous rocks of St David's Head and those lying further east, especially at Pen Berry, appear to be somewhat moutonne on their northern aspect, but no unmistakable glacial striae were seen, and this is no doubt due to the fact that the rock-faces are so much weathered. On the greater heights there is a general absence of perched blocks and big erratics. It is quite possible that perched blocks may have been common in former times, but they have in all probability been removed by man, for the region is full of traces of defences prepared by primitive man, and these usually take the form of great collections of boulders and stones gathered together and heaped up in the form of dykes. Splendid examples of such ancient entrenchments are seen near St David's Head, and again on Strumble Head. The few blocks seen on the high ridges are almost invariably of the same nature as the underlying and surrounding rocks. South-west of St David's, in the Treginnis tract, some huge boulders are seen. A big one lies on the hill above the cliffs at Penmaen-melyn, but it consists of a somewhat coarse andesitic rock, which is found in situ at no great distance away. The boulders of granite and of picrite found near Porth-lisky farm by Dr Hicks have been already mentioned. The whole country was once strewn with boulders ; and although many still remain scattered over the land, most have been cleared away and used for building dykes, etc. An examination of the stones in the dykes shows that they are almost entirely of rocks found in the locality, as might be expected. Boulders of the St David's Head gabbro are found carried in a south-eastern direction, and are plentiful in THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 63 the neighbourhood of Caerfai Bay and Caer Bwdy Bay. In the country between St David's and Strumble Head boulders are also common. Blocks of "greenstone" are found around Mathry : the rock is a diabase, probably occurring at no great distance away. On Strumble Head the boulders are very plentiful, and the writer found a boulder of picrite not far from the extreme head, near the coastguard station, and another further south, near Tre-Seissyllt, together with a remarkably fresh olivine-gabbro, of a kind which is not found in the district. These will be referred to more fully in another section of the paper. East and south of Fishguard, boulders of a dark "greenstone" and of a volcanic rock which weathers white are abundant. They are found on the hill- slopes, on roadsides, and in the fields, but many have been cleared away as before. The Precelly hills are free of boulders as compared with the hills lying further north. It may be mentioned that vestiges of antiquity abound in this part of Pembroke- shire. Hut circles, ancient entrenchments, cromlechs, British camps, etc. occur at places all the way from St David's Head to Cardigan, being particularly evident at St David's Head and Strumble Head. Some of the ancient British towns or settlements, traditions of which are to be found in the old Welsh romances, called The Mabinoonon, are situated within this area, and one has been identified as occurring on the Garn Fawr, Strumble Head. IV. Description of the Deposits. The glacial deposits of the district vary a great deal as traced laterally from place to place. Owing to the want of good exposures inland, it is generally impossible to mark out the limits of the different kinds of superficial detritus. One has to depend for most part on a study of the sand-pits and clay- or marl-pits which are dug in places all over the district. But it is not often that these go down to any great depth ; and when occasionally a deep pit has been dug out for the purpose of obtaining clay or sand for the land or for building, it is invariably filled up again with surface- rubbish, so as not to be a danger to animals. Where a pit appeared to be of peculiar interest, the writer employed a man for digging, and in this way obtained some valuable sections. At the eastern limit near Cardigan there are brickworks, and here it is that the best sections are to be seen. The writer hopes in the near future to con- tinue his investigations eastwards in the neighbourhood of Cardigan. The deposits, which occur in the district are the following : — 3. Upper Boulder-Clay and Rubbly-Drift. 2. Sands and Gravels. 1. Lower Boulder-Clay. 1. The Lower Boulder- Clay. — This is a typical boulder-clay which is met with in patches throughout the district, but is best and most fully developed towards the east. It has received no attention within recent times, but a very quaint and, on the whole, a very accurate description of it is found in the works of a writer who lived in <U DR T. J. JEHU ON the time of Shakespeare. In an article on Sir Roderick Murchison's Silurian System in the Edinburgh Review, 1841 (vol. lxxiii. p. 3), it is stated that " one of the oldest inquirers connected with the geology of this ancient region is George Owen of Henllys, in Pembrokeshire, who has been called the patriarch of English geologists." This worth}- Welshman left behind him a manuscript work on the topography of his native country — a book of great value and interest. It was published in the Cambrian Register, 1793, and has recently been reproduced, under the editorship of Mr Henry Owen, in the Cymmrodorion Record Series. The book has been already referred to, and is entitled The Description of Pembrokeshire. His observations on the boulder-clay are so good that they are well worth quoting. Writing of " the naturall helpe and amendementes the soil it selfe yealdeth, for betteringe and mendinge the lande," he refers to what he calls " Claye Marie." " This kind of Marie is digged out of the Earthe, where it is found in great quantitie, and thought to be in rounde great heapes and lompes of Erthe as bigg as round hills, and is of nature fatt, toughe, and Clamye. . . . The opinion of the Countrie people where this Marie is founde is that it is the fattness of the Earthe gathered at Noes flood, when the Erthe was Covered withe the said flood a whole yeare, and the surginge and tossinge of the said flood, the fattness of the Earth being clamye and slymie of nature did gather together, and by rowlinge vpon the Earthe became round in forme, and when the flood departed from the face of the earthe, the same was left drie in sondrie partes, which is nowe this Marie that is found, and how the Common people Cam to this opinion I knowe not, but it is verye like to be true, for wheresoever the same is founde, it is loppie (loose) and covered with sande, gravell, and round peblestones, such as you shall flncle at the sea side verie plaine, appearing that the stones hath ben worne by the sea or some swift river." " Also in the harte of the Marie is founde diverse sortes of shells, of fishe, as Cogle shells, Muskell shells, and such like, some altogether rotten & some yet unrotted, as also you shall therein finde peaces of tymber that ben hewen with edge tools & fire brandes, the one ende burned and diverse other thinges which hath ben before tyme vsed, & this XX tie foote and more deepe in the Earth in places that never haue been digged before, and over the which great oakes are now growinge ; and this seaven or eight myles from the sea, so that it is verie probable that the same came into these places at the tyme of the great and generall flood " " This marie is of couler with vs most commonlie blwe and in some place redd." " It is verie hard to digg by reason of the toughness, much like to waxe : and the pickax or mattock beinge stroken into it, is hardlie drawne out againe, so fast is it holden : it is alsoe verie heavie as ledd." " This Marie is founde in Kernes and both Emlyns from Dynas vpp to Penboyr in Carmerthen sheere, beinge about twentie myles in lengthe and about fowre myles in bredeth in most places to the sea syde, and out of this compasse I cannot heare that the same ys founde : I thinke more for want of Industrie than otherwise " (pp. 71, 73). He ends up his remarks on the Clay Marie THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 65 thus : — "And, who so list to learne more of this Marie : let him pervse a pamphlett which I have written thereof, wherein I have declared the nature of the marie, how to know yt and finde yt, and the order at Lardge of digginge and layeinge yt on the lande ; of the severall sortes thereof for what yt is good, and for what yll. And so for brevyties sake I Cesse to writte any More thereof." It is a great pity that this pamphlet has never been published, for it would be of great interest to geologists, as perhaps the earliest attempt to give a full description of the boulder-clay. It seems that the treatise was finished in 1577, and consists of twelve chapters. A footnote to the preface of Mr Henry Owen's edition of his Pembrokeshire (p. xxiv) states that a copy of the pamphlet lies in the Vairdre Book at Bronwydd, " written out of a copy in his own hand, by me, John Owen of Berllan, 1684." The Kernes mentioned in the above extract is that part of Pembrokeshire lying between Dinas and Cardigan. The present writer can bear testimony to the general accuracy of the description given by George Owen. It is at the brickworks, Cardigan, that the best exposure is seen. There in the pit a vertical section of this boulder-clay 20 feet deep is seen. It is dark-bluish in colour, but after drying becomes more of a light bluish-gray. It is a remarkably tough and tenacious clay, and can only be dug with great difficulty, for no crevices or fissures are seen and no trace of bedding. The whole mass is strikingly homogeneous and uniform in character, and has evidently been subjected to great pressure. The bottom is not reached in the section, and so the depth attained by it at this place is not known. For most part it is very free from stones, but a little further east in the same pit these are rather more commonly met with. Some beautifully glaciated sub-angular and blunted boulders were seen, with the striae running princi- pally in the direction of their longer axes. Many of these boulders are of Carboniferous Limestone, and these interfere very much with the manufacture of the bricks, and are, as far as possible, picked out by the workmen. Boulders of conglomerate, grit, shaly and slaty rocks were also noted, and many of igneous rocks, which are foreign to the district. These will be dealt with again below, in another section. One of the most characteristic features of this Lower Boulder-Clay is the presence of marine shells scattered irregularly through it. They seem to occur chiefly in its upper part, and are invariably much broken and worn, and therefore very difficult to identify. The fragments are also extremely friable. Occasionally small waterworn pebbles of quartz, etc. are seen in the clay ; but most of the stones included are ice- worn rather than waterworn in the Cardigan pit. Another striking feature is the presence of fragments of woody matter in the clay, sometimes at a depth of 15 to 18 feet. Above the boulder-clay in the brickyard occurs 2 or 3 feet of sand and gravel and a yellowish stony clay, and towards its north end this stony clay increases in thickness to at least 7 feet, passing in places into yellow sand. 66 DR T. J. JEHU ON At Cardigan the height of the boulder-clay above sea-level is under 50 feet. Between Cardigan and Dinas this blue clay is seen in patches underlying small tracts of moorland, and it attains a height of nearly 600 feet a little south of Pen-Creigau, where, at a short distance below the road to Cardigan, it may be seen, though the exposures are very poor. Just south-west of Dinas, near the roadside, clay-pits occur on Rhos-Isaf showing a depth of 6 feet. The same stiff, compact, bluish boulder-clay is here seen, full of com- minuted shell-fragments. Boulders are fairly common, mostly ice-worn and scratched, but some water- worn. One example of a slaty rock showed not only fine striae but a wide groove smoothed out by ice action. The clay gets darker as traced downwards, but the bottom is not seen. Workmen stated that it reaches a depth of at least 1 5 feet, and occasionally a thin seam or stratum — no more than half an inch in diameter — of fine gravel is said to occur. But no trace of bedding occurs in the clay. It is capped for 2 feet by a yellowish clay with boulders. The pits are 240 feet above sea-level. Small exposures are seen in some of the fields on Dyffryn farm, about a mile south of Goodwick. Owing to drying and weathering, it is of a light bluish-gray colour, and is here full of fragments of the local lavas and tuffs. Most of this farm is underlain by this clay. A little further south, in a field belonging to Drim farm, is a small pit of a similar character. No shell fragments were to be seen in these exposures. Similar tough bluish boulder-clay is seen in clay-pits on Tregroes moor, at a height of over 250 feet. In fact, in all the moors lying to the south of Fishguard, with the exception of those found on mountain-slopes, this boulder-clay can be found, but it is unfortunate that there are no deep pits or good sections to be seen. But the engineers of the Great Western Railway are making borings in this neighbourhood for a tunnel, and they very kindly supplied the writer with all the information in their possession which might be of interest. A boring has been made at Trebrython farm, to the south-west of Tregroes moor, and about 150 yards from the railway. In the boring the following succession was obtained : — 5 feet of earthy clay. 5 feet of yellowish clay, with rock fragments. 10 feet of a somewhat tough greyish-blue clay. Slate rock. The greyish-blue clay is the Lower Boulder-Clay, and it is seen that it only attains a thickness of 1 feet here. The yellowish clay may be partly the bluish clay weathered, but most probably consists for most part of the equivalent of the Upper Boulder-Clay. The boring is made at a height of nearly 300 feet above sea-level. About three- quarters of a mile down the railway towards Goodwick is a railway cutting passing through part of the superficial deposits, and a boring has been made here also. A full section of this cutting and boring is given on another page, but it may be mentioned here that typical tough blue boulder-clay is there shown which attains a depth of THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 67 18 feet, and is followed above by Sand and Upper Boulder-Clay. The Lower Boulder- Clay is partly exposed in the cutting, and is full of fragments of marine shells. There is also a small exposure in the moor just south of Letterston. That the boulder -clay occurs far inland is proved by an examination of a pit at Llyn, near Llangolman, not far from Maenclochog. When seen the pit was only 5 feet deep, but it is opened up by the farmers periodically to a depth of from 15 to 20 feet, and ladders have to be used to get in and out. The clay is bluish in colour and very tough, and is usually spoken of by the natives as " indiarubber clay." It occurs in boggy land at the bottom of a small valley, at a level of over 400 feet. On both sides the land rises to a higher level, and is covered by a mantle of sands and gravels. West of Goodwick the blue clay comes to the surface just below St Nicholas Church, at Clyn Bach moor. The depth was not ascertained. Fragments of broken shells occur here, though somewhat rarely. Some boulders of igneous rocks, foreign to the district, were found, and among them an unmistakable boulder of the Ailsa Craig riebeckite micro-granite. The bluish clay is seen at places as far as the western end of the promontory — such as in the moors around Trefain, and in the shallow pits at Henllys, below Llanrian, where very fine examples of chalk flints were seen included. In the pits of Trefeithan moor, west of St David's, this boulder-clay is exposed to a depth of 6 feet and is of the usual character. No shells were seen, but some vegetable matter occurs in the clay. The boulders were few, and those noted were all of local rocks. Somewhat similar but shallower pits may be seen on Dowrog moor, Tretio moor, and Caer-farchell moor. It is thus evident that this Lower Boulder-Clay occurs throughout the district, but it appears to thicken as we pass from west to east, and to be best developed to the east of Strumble Head. Fragments of marine shells were found in the clay at Cardigan, at Dinas, at the railway cutting near Tregroes, and at St Nicholas, and for the deter- mination of these and other shells mentioned in this paper the writer is indebted to Mr Henry Woods, M.A., St John's College, Cambridge. Owing to their fragmentary condition it has been difficult to identify with certainty the shells found in the clay. But the following species appear to be represented : — Cardigan Clay-pit — Pectunculus glycimeris, L. Astarte sulcata, Da Costa. Mytilus, sp. Dinas Clay-pit — Astarte (Nicania) compressa, Mont. (?) Cyprina islandica, L. 68 DR T. J. JEHU ON Railway Cutting (between Tregroes moor and Manor-owen) — Pectunculus glycimeris, L. Cardium islandicum 1 Chem. Vulsella modiola 1 L. Astarte sulcata, Da Costa. Astarte (Nicania) compressa, Mont. Venus (Ventricola) carina, L. Cyprina islandicu, L. Some of the boulder-clay from the boring near Tregroes was washed and examined under the microscope. A good deal of very fine sandy material — mostly quartz — was observed, and a few foraminifera could be seen. 2. The Sands and Gravels. — Above the Lower Boulder-Clay comes a series of aqueous deposits, consisting of sands and gravels, which are sometimes stratified and sometimes show hardly any traces of stratification. These deposits vary very much in thickness, and are apt to die out suddenly when traced laterally. They usually occupy a higher level than that attained by the Lower Boulder-Clay, and are often seen banked on the lower slopes of the hills. In places where sections are seen passing through the different deposits, no gradual passage can be traced from the lower stiff blue clay to the sands and gravels above — and this suggests that the sands and gravels lie upon an eroded surface of the clay. The sands are as a rule yellowish and yellowish-brown in colour, and have all the appearance of being marine : they are very variable in texture, and show all gradations from very fine sand to coarse gritty sand and gravel. And the gravels are often coarse and pebbly, resembling the shingle collected on beaches. At some places the sands are charged with worn and broken fragments of marine shells : these usually occur more abundantly in the fine gravel or coarse gritty sand than in the fine sand, although minute flakes can often be detected in the latter. They seem to be collected together in the stratified beds at certain spots, and to be absent in somewhat similar beds exposed only a short distance away. For instance, shells are plentiful in the Manor-owen sand-pit, whilst not a trace of shells can be seen in the Cnuc sandy pit, which lies only about a quarter of a mile further south : and 50 yards or so beyond the Cnuc sandy pit shell-fragments are again found in sand exposed in the railway cutting. In the pits where shell-fragments are found it may often be noticed that small pieces are cemented to the surface of a rounded stone. This is doubtless due to a deposit of carbonate of lime derived from the decay of some of the shells. Like the Lower Boulder-Clay, the deposits of sand and gravel become better developed when traced from west to east, and the best sections and pits can be seen in that part of the district lying to the east of Strumble Head. Though the St David's promontory is largely covered by a loamy sand, no sections were seen showing deposits of the well-marked marine-like sands and gravels found further east. At Ty-llwyd there are small pits reaching a depth of 5 to 6 feet, where the loamy sand is well THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 69 shown. The bottom of the deposit is not seen, but as traced downwards the material becomes very sandy. It is ferruginous, and has a reddish colour. The sands and gravels become more evident in the neighbourhood of Mathry. A little to the east lies a rugged mass of igneous rock known as Y Graig, and on the south side of this there is a newly dug pit, exposing 8 feet of pebbly gravel and gritty sand, with streaks of fine yellow sand. The deposit gets more sandy as traced downwards, but the bottom is not seen. A tendency to a rough bedding is shown in the section. At Pont Duan, north of the roadside, a very similar gravel-pit is seen. Pen Cnuc, at Castle Morris, marks the site of a mound of fine yellow sand, most of which has now been carried away. Gravel and sand is exposed, but only to a depth of 4 feet, in a field 200 yards north of Bridge-end, and again at Heathfield, south of the house. Further north, Tre-gwynt lies on sand and gravel, and much sand is seen between Tre-gwynt and Trellys. East of St Nicholas Church the sands and gravels cover most of the land, as may be seen in pits in many of the fields. The writer employed a man to dig here in order to ascertain if possible the depth attained by the sand and gravel. At 8 feet the bottom was not reached, but the gravel became wet, and it is probable that the blue clay lies a few feet lower down, for it crops out in the moor below St Nicholas Church. The pebbles in the gravel-pit were all well rounded, and chalk-flints were seen. Gravels and sands seem to cover much of the ground on the Strumble Head promontory, but at places a yellowish earthy clay replaces them at the surface, though they may here occur with the clay. In a boring made for a well at Llandruidion farm sand was brought to the surface, in which comminuted shells were seen. The boring reached a depth of over 20 feet, and rock was not reached. In the farmyard at Tre-howell, near the northern extremity of Strumble Head, in sinking for a well, no sand was passed through — all was earthy clay ; but in a field 250 yards further north a sand-pit occurs, where 2 feet of fine yellowish sand are seen, covered by 2 or 3 feet of a loamy and somewhat stony clay. Mounds of gravel and sand occur on Caergowil, on the heights above Goodwick, and sections 7 feet deep are exposed. They are of the usual character. South and east of Goodwick and Fishguard deposits of sand and of gravel are frequently met with. They do not occur in the form of kames or eskers, but are found lying in the slopes of the minor hills, and sometimes spreading to the top. It would be almost impossible to map them, as their occurrence is so irregular and patchy ; they are apt to die out laterally in a sudden way, passing into clay or rubbly-drift. At many places they are overlain by an Upper Boulder - Clay. Perhaps the most interesting of all the sections is one seen in the Manorowen sand - pit, which lies in a small wood on the roadside, immediately south of the farm buildings, at a height of nearly 200 feet above sea-level, and two miles distant from Goodwick Bay. When visited a section of only 5 feet was seen, but means were taken to deepen it down to 12 feet. A somewhat diagrammatic view of this section is shown in fig. 1. TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 4). 12 70 DR T. J. JEHU ON The materials were well stratified, and occur in the following order from above downwards : — A. Soil and rubbly-drift 1-3 feet. B. Very fine light-brown sand, with the beds somewhat contorted, passing down into thin beds of darker sand, followed by coarse sand with some pebbles. Near the bottom is a thin band of sandy clay, 3 inches . . . . 4£ feet. C. Pebbly sand, variable in thickness . . . . . . . 1 to 1J feet. D. Coarse grey sand or line gravel, showing bedding. Some layers are more distinctly pebbly, and here and there fine sand occurs. The fine gravel is full of fragments of marine shells ........ 6 feet. Bottom not reached. 1-llt , u w j*ittobd**~ Jil '**«*ta aiJ _ Soil and rubbly- drift. Fine sand, show- ing some fold, iug. Sand with peb- bles. ~ - — --«. — -»- Pebbly sand. Coarse grey sand and fine gravel with shell frag- ments. Greatest length 1 2 feet. Depth about 12 feet. Fig. 1. — Diagrammatic Section of the Manorowen Sand-pit. Many of the shells have been identified, and are discussed below. Chalk-flints are common. On the opposite side of the road, below the churchyard, and at a lower level, there is a small exposure — 3 to 4 feet deep — which consists entirely of a coarse gravel ; but above, towards Manorowen Hill, the gravels and sand are replaced at the surface by clay. About a quarter of a mile further south, sand is seen again at Cnuc Sandy. There is a big pit just in front of the cottage, 8 feet deep. What is seen here is for most part very fine yellow sand. Gritty and gravelly streaks and layers occur here and there, dying out as traced horizontally. No traces of shells were found here. The pit was at THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 71 one time 15 feet deep, and the bottom of the sand was not then reached. 1 to 3 feet of stony-drift cap the sandy beds at the surface. At a distance of 50 yards further south the railway cutting has passed through 1 feet of similar very loose sand, in which fragments of marine shells occur plentifully. The town of Fishguard is, in part at any rate, built on sandy deposits, and a good exposure is seen in a quarry on the roadside going down Fishguard Hill to Goodwick Bay. It consists of yellowish sand and fine gritty gravel of the usual kind, which near the top becomes more of a loamy, stony drift. North of the town gravel- pits are common in the fields, and at Pwll Landdu on the coast, east of Castle Point, the cliff is largely made up of gravel and a ferruginous sand, capped by a yellowish boulder-clay full of stones. A little north of the valley of the Gwaen, at Tre-llan, near Llanllawer, fine yellow sand occurs on the lower slope of the hill of Ceunant. It would be useless to mention every spot where the sands and gravels are to be seen. They occur in patches all the way to Cardigan, being especially well seen in Llwyn-y-Gwaer Park. The highest level attained in this part of Pembrokeshire by the sands and gravels is at Pen Creigiau Cemmaes, just off the road leading from Nevern to Cardigan, and four miles distant from the latter place. Sand occurs at the top of the hill, at an elevation of 640 feet. Most of the hill-top is evidently of sandy material, and in a pit a section 8 feet deep is seen, showing very fine yellowish sand passing downwards into darker and more gritty material. There is only a faint trace of bedding. The bottom is not seen. This spot is nearly three miles distant from the coast. Chalk-flints and well-rounded pebbles of quartz are found. A few yards down the hill on the northern side are other small exposures, about 4 feet deep, showing more pebbly sand with rounded boulders ; and on the southern side, immediately below the main road to Cardigan, is a gravel-pit, in which are seen rounded and sub-angular stones, some a foot in length. Chalk-flints and pebbles of white quartz were common, and a boulder of Millstone Grit and of a reddish granitoid rock foreign to the district were picked up. Also two pebbles of a muscovite granite. These will be referred to again below. A rough kind of stratification could be seen — layers of small gravelly pebbles separating beds of coarse shingle. The pit is 8 to 10 feet deep. At Pant-gwyn, half a mile north of Pen Creigiau, sand is seen in a pit, and it is darker and more gritty than that on the hill-top. Deposits of material resembling marine sands are met with even north of Cardigan, as at Banc-y-warren, but this is outside the area embraced in this paper. Similar deposits are found far inland, even south of the Precelly hills. A few yards south of Eose-bush there is a sand-pit on the western side of the railway. A diagrammatic section of it is shown in fig. 2. The lower part is hidden by a talus slope. Above this comes 4 to 5 feet of fine yellow sand, very ferruginous in places. The sand becomes a little clayey or loamy in the eastern half of the section, and is 3-4 feet - - 4-6 feet • — rHS 4-5 feet 72 DR T. J. JEHU ON traversed by patches and imperfect layers of a blackish hard pan-like material, which is probably organic in nature. Above this comes 3 to 4 feet of rubbly material, full of fragments of slaty and other rocks of local origin. Fine yellow sand is seen also near Llangolman, a little east of Maenclochog, on the slopes of the ground rising from the moor, and at Charing Cross there is a pit showing roughly-bedded gravel and sand to a depth of 8 feet. A smaller pit of a similar nature lies at Cefn Ithyn, just north of Maenclochog. Further south, in the neighbourhood of Trefgarn, sand of quite a different character occurs. There is an exposure of 12 to 15 feet in a big gravel-pit on the roadside opposite the Chapel-of-ease, near Nant-y-coe mill. The fine gravel and sand here is dark grey in colour, and consists largely of minute flattened flakes. The sand is not ^ ^ ,» J » J ..» r .^^»<r. ^3'g*r^33g5 S3aB».^^ Soil. Iff 1- !*. ^^^5?^&^^^-~ Fig. 2. — Diagrammatic Section of the Rose-bush Sand-pit. yellow nor brown like that found further north, which the natives speak of as " Demerara-sugar " sand. In Trefgarn Hall park there is a pit where the material seen is somewhat similar, but much coarser. The stones are rounded and sub-angular. The deposits here do not remind one so much of the sand and gravel found on sea- shores. List of Mollusca found in the Manorowen Sand and Gravel Pit* Lamellibranchia. Nuculana pemula, Mull. ......... rare. Pectunculus glycimeris, L. ........ . very common. Barbatia lactea, L. ......... . very rare. Mytilus edulis, L. .......... rare. Vulsella modiola 1 L. . . . . . . . . . very rare. Astarte sulcata, Da Costa ......... very common. Astarte (Nicania) compressa, Mont. ....... moderately rare. Astarte (Tridonta) arctica, Gray ( = boreal 'is) ...... moderately rare. * The specimens which I collected from this place, together with some obtained subsequently by Mr V. M. Turnbull, are now in the Sedgwick Museum, Cambridge. THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 73 Cyprina islandica, L. . Tellina (Macoma) balthica, L. Maetra (Spisula) solida, L. . Venus ( Ventricola) easina, L. Tapes {Amygdala) decussatus 1 L. Cardium islandicuml Chem. Mya truncata, L. . common, moderately rare, rare. moderately rare, very rare, very rare, rare. ScAPHOPODA. Dentalium entalis 1 L. . Gasteropoda. Puncturella noachina, L. very rare. Natica clausa 1 Brod. and Sow. . rare. Turritella communis, Lam. common. Buccinum undatum, L. common. Tritonofusus gracilis, Da Costa very rare. Ocinebra erinacea, L- very rare. Trophon (Boreotrophon) clathratus, L. . common. „ „ scalar if or mis, Gould moderately rare Nassa (Hima) incrassata, Strom. . very rare. Bela turricula, Mont. rare. Bela rufa, Mont, (elongate form) rare. Fragments of other shells were also found, but they were too broken for identifica- tion. Some of the shell fragments found were very thick, especially pieces of Cyprina islandica. Many are rolled, and the majority very broken. Entire single valves of Astarte compressa were found ; and of the univalve shells, Ocinebra erinacea and Trophon clathratus occurred in nearly perfect condition. The fauna appears to contain a mixture of species belonging to different climates : Astarte borealis, Trophon clathratus, and Trophon scalariformis are Arctic and Scandinavian species, not now found living in British seas. Astarte compressa, Cyprina islandica, Buccinum undatum, and Puncturella noachina belong to a northern type of British species which inhabit Arctic and Scandinavian seas in common with our own. The shells are in very much the same condition as those which have been obtained at Moel Tryfan and at Gloppa, and most of the forms found at Manorowen occur in the other two places also. But Nuculana, which occurs rarely at Manorowen, is common at both of the other places. Pectunculus glycimeris is abundant at Manorowen, but very rare at Moel Tryfan and Gloppa. Venus easina, though frequent at Manorowen, is also rare at the other places. Samples of the sand from several places were examined microscopically, and they all showed a very close resemblance to marine sands. Most of the grains were of 74 DR T. J. JEHU ON quartz, and the smaller ones were angular, while the bigger ones tended to be more rounded. 3. TJ'p'per Boulder-Clay and Rubbly-Drift. — The sands and gravels are in many places covered by a yellowish-brown boulder-clay, quite different in character from the bluish boulder-clay which underlies them. This Upper Boulder- Clay is sometimes fairly tough, and is generally much more stony than Lower Boulder- Clay. It varies very much in thickness and character. Inland it often only occurs as a thin covering a few feet deep, but on the coast, where the best exposures are seen, much greater depths are attained. Sometimes it is a tumultuous unstratifled till, with boulders of all shapes and sizes scattered pell-mell throughout the matrix. At other places — and it may be at no great distance away — it has more of the character of a rubble-drift, and as seen in section, has the appearance of an agglomeration of coarse and more or less angular debris, showing a rude kind of bedding. It is evident that most of it consists of material which has been re-arranged to some extent, and afterwards modified by sub-aerial agencies. It is impossible to separate the more typical unstratified boulder-clay from the rough semi- stratified clayey and sandy rubble-drift. The included boulders are derived in the main from the rocks of the district, but many far-travelled stones are also found, and these will be discussed in the next section. Ice-scratched stones are fairly common. These are usually sub-angular, with blunted angles and rounded edges. Bounded waterworn stones are also common, especially in the re-sorted rubbly-drift. No traces of marine shells are seen in the Upper Boulder-Clay and Rubbly-Drift. On the coast it is found capping the rocky cliffs at places, and in the bays fine sections, sometimes over 20 feet deep, are exposed. The foreshore is often covered with big boulders derived from the neighbouring cliffs. As the sea is now gradually gaining on the land, the cliffs of drift on the coast are being continually undermined, and the included stones and boulders are washed out and strewn over the shores. The beaches on the bays are rich in boulders and stones of rocks foreign to the district. These have undoubtedly been derived from the cliffs of drift, which are constantly undergoing a process of degradation owing to the action of the waves and of sub-aerial agencies. By far the best exposures of these upper deposits are shown on the coast-line of the St David's promontory. Figs. 4 and 5 (Plate) represent sections seen in Whitesand Bay, and give a very good general idea of the appearance of these Upper Drift deposits. Fig. 4 shows a section of the cliff near the north end of the bay. Here the cliff is about 20 feet high and consists entirely of drift. At this spot the drift is a typical till or boulder-clay, showing no bedding, but full of stones and boulders, big and small, which are scattered confusedly through it. Most of the boulders are angular and sub-angular, and some are well glaciated. A few rounded pebbles and stones also occur. The boulders are mostly of local rocks, though some erratics are seen. Loose sandy soil occurs at the top. At the base the rock does not appear, but the shore is covered with stones derived from the cliff. A few yards further north slaty rock is seen THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 75 emerging from underneath the boulder-clay, which becomes thinner in this direction. At places the rock shows a hummocky surface, marked with glacial strise, which run from north-west to south-east. At the base of the small promontory called Trwyn Hwrddyn, on the north side, a rubble of very coarse fragments lies between the solid rock and the drift. Here the drift shows a rough sort of stratification, and has much sandy and pebbly material intermingled with boulder-clay. Fig. 5 shows a section in Whitesand Bay seen further south. This is also about 20 feet in height, but here it consists more of a rubbly- drift. The matrix is earthy and sandy, and is choke-full of small flakes of slaty and other rocks, which have a rude kind of arrangement, especially towards the lower part. Boulders of larger size occur here and there, and consist for most part of grit and conglomerate and slaty rocks, with some blocks of the local gabbros and diabases. Pebbles of white and yellowish- white quartz are common. The top is covered by loose yellow sand, probably wind- blown, and just underneath this are traces of a pebbly bed. Though rock does not appear at the bottom just at this spot, it crops out on both sides a short distance away. Drift of a similar kind is seen in sections, and capping the cliffs at other places on the western coast of the promontory. Boulder-clay is seen at Porth-lisky stuck full of stones, many of which are smoothed, polished, and striated ; and a boulder of the St David's Head gabbro, measuring roughly 3x2x2 feet, lies on the beach below. A stony boulder-clay or a more rubbly drift is seen at various places on the south coast, the best exposures being above Caerfai Bay and at Caerbwdi Bay. At Caerbwdi the cliff is over 20 feet in height, but the base is hidden by talus : the matrix is here rather sandy, and streaks and pockets of rather fine sand are seen here and there. The included stones are often pebbly, but some are sub-angular and ice-marked. They are made up almost entirely of rocks found in the neighbourhood. Near the top flaky fragments are very plentiful, and these are derived from the local purple flagstones and slates. A big boulder of the coarse gabbro from St David's Head lies at the base of the cliff. No good section is seen at Porth-y-Bhaw, but the drift caps the hills and cliffs to the south-east. Very similar sections are seen on the north coast at Abereiddy and above Traeth Llyfn. At the latter place a rubble of big boulders is seen ; most of these are of local igneous rock, very iron-stained and decomposed. Towards the top the section is freer of big boulders, and is full of little flakes of sedimentary and cleaved rocks. One of the finest sections on the coast is seen at Aber-mawr, west of Strumble Head. At the northern end the rock is seen capped by 10 to 15 feet of stony-drift. As traced southwards the drift thickens to about 40 feet, then tails off rapidly. Where thickest the lower part shows some tendency to a rough kind of bedding, and is full of small flakes and little stones, more or less pebbly. This passes above into a rubbly clay, full of boulders of all sizes, most of which are angular and sub-angular, and derived from rocks of the locality. The cliffs on Strumble Head in many places are covered by a 76 DR T. J. JEHU ON mantle of stony till, but good sections are rare. The sections of drift exposed on the coast between Strumble Head and Cardigan are not so good as those seen west of Strumble Head, but where seen they are of the usual character. Inland, good sections are not often met with. Much of the country in the neighbourhood of St David's is covered by drift, which consists of a sandy matrix full of boulders. But often the matrix is clayey, as may be seen in sections on the roadside near Castell, south-west of St David's, and again in a pit just off the road near Pont Clegyr, two miles east of St David's. In fact a large part of the country south-west of Strumble Head is covered by material which has been to a large extent re-arranged, and which cannot be defined accurately either as boulder-clay or as sand and gravel, though the tendency is for the sand and gravel to become more marked at a depth of a few feet from the surface. Much rubbly-drift, becoming more sandy as traced downwards, is spread out on Strumble Head, especially on the moorland above Goodwick. The sands and gravels occurring south and east of Fishguard Bay are, as already mentioned, usually overlain by a few feet of rubbly-drift or more typical stony boulder-clay. In the railway cutting between Tregroes moor and Manorowen 7 feet of stiff yellowish-brown boulder-clay is seen covering the shelly sand. This clay is spread out over much of the high land skirting the railway here on the west side. A little further south the sands and gravel die out, and the Upper Boulder-Clay seems to lie directly on the Lower Boulder-Clay, and this occurs possibly in the boring at Tre-bython already referred to, where yellowish clay is succeeded by tough bluish clay. It is very rarely that one has an opportunity of finding all the deposits succeeding one another in the same section, and of ascertaining the depth of each. But the engineer of the Great Western Railway at Goodwick supplied the author with particulars of the boring made in the railway cutting between Tregroes Moor and Manorowen, just about Cnuc Sandy. A complete section of the railway cutting, together with the results obtained by boring, are given on the next page. From above downwards, the deposits passed through were — (4) Stiff yellowish-brown clay with fragments of slate-rock .... 7 feet (3) Fine yellow sand with shell-fragments . . . . . . 10 „ (2) Stiff dark-blue boulder-clay with shell-fragments . . . . 18 „ (1) Gravel 5 „ Rock. This section proves very clearly the presence of an Upper and a Lower Boulder-Clay. These are separated here, as in many other places, by a deposit of sand. Most of the Lower Boulder- Clay is below the surface of the railway cutting. And a glance at the section shows that before the cutting was made, the surface of the ground was covered to a depth of 7 feet by the Upper Boulder-Clay. So it is quite possible that the Lower Boulder-Clay spreads over a much wider area than is evident at the surface, and that much of it is hidden by more superficial deposits. It is interesting to note that underneath the Lower Boulder-Clay there lies 5 feet of gravel. On comparison this was THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 77 found to be very similar to underlying grit rock when this is broken up. So it may represent material ground out of the solid rock by the movement of land-ice. Further inland a good example of the Upper Drift is seen in a cutting on the roadside between Puncheston village and the railway station. It is a stony and rather rubbly clay. Close at hand, on Puncheston common, the Lower Boulder-Clay crops to the surface. Further east, up to Cardigan, the Upper Boulder-Clay and Rubbly -Drift are seen in many places overlying sands and gravels or lying immediately on the blue clay. The line of division between this Upper Boulder-Clay and the sands and gravels is not so marked as that between the sands and gravels and the Lower Boulder-Clay. Stole AOff to \iq . Fig. 3. — Section at Boring No. 1 on Goodwick to Letterston (existing) Railway, between Tregroes Moor and Manorowen. V. The Boulders and Erratics. The transport of boulders is of great importance as indicating the general direction of ice-movement. Throughout northern Pembrokeshire boulders may be seen scattered over the surface, and are especially common on waste or uncultivated land. Many of those found on the slopes of the hills have no doubt rolled from the parent rock above, but many, though not foreign to the district, have yet been carried for some distance by ice. Where the land is cultivated an immense number of boulders must have been removed, but even in the fields it is common to meet with big blocks of igneous rocks, which, as a rule, have been left in the position where they were originally stranded in order that cattle may have something to rub against, or have been left on account TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 4). 13 7S DR T. J. JEHU ON of their immense size, though these latter are becoming gradually destroyed by blasting operations. At many places, especially on the St David's plateau, huge standing stones, cromlechs, or other ancient remains are seen, and it is more reasonable to believe that the immense blocks used for these purposes were found as boulders near at hand than that they were quarried from the parent mass, which often lies at a considerable distance away. And this is rendered the more probable as blocks of similar size are not infrequently seen dotted over the surface. An examination of the stone dykes will show how plentifully boulders of all sizes must at one time have been studded over the ground, and what a variety of rocks is represented. Towards the western end of the area the boulders are largely found to have been dispersed from the igneous rocks on the north coast of the promontory. And everywhere the great majority are derived from parent masses found in the district. Boulders of diabase may often be seen resting on volcanic or sedimentary rocks, and vice versa, proving that there has been some transport. It has already been mentioned that blocks of the St David's Head gabbro are found lying to the south-east at Caerfai Bay, and Caerbwdi Bay, and on the cliffs above. This implies that there was a movement of ice from a north-westerly direction, and it agrees with the evidence shown by the glacial strise which are seen on the coast. Erratics are met with often in the drift and on the shores, but the number which has been noted on the surface is not great. A further study of the stone dykes would doubtless bring more to light 1. Erratics seen on the surface of the ground. — The detection of those about to be mentioned is mostly due to the fact that they had been broken up by the farmers through blasting or other agency, so as to expose fresh surfaces. Hicks states (Geol. Mag., 1891, p. 501) that he observed many northern erratics in the St David's district. The granite boulder which he discovered on Porth-lisky farm " before it was broken must have been over 7 feet in length and 3 to 4 feet in thickness, and is identical with a porphyritic granite exposed in Anglesea." He found another of picrite which he thus describes : " The boulder is somewhat rounded ; its longer axis, which lies nearly south- east and north-west, measures about a yard. A transverse section is slightly triangular, the shorter sides measuring respectively about 16 inches and 22 inches. It lies on the promontory forming the east side of Porth-lisky harbour, resting immediately on Dimetian rock, surrounded by an uncultivated area overgrown by gorse and heather " (Quart. Journ. Geol. Soc, vol. xli. p. 519). It was submitted to Professor Bonney for examination, and he states that it is wonderfully like the boulders found at Pen-y- Carnisiog, Anglesea, which had been previously described by him (Quart. Journ. Geol. Soc, vol. xxxvii., 1881, p. 137). In a later paper (Quart. Journ. Geol. Soc, vol. xli., 1885, p. 518), Professor Bonney remarks that " the lithological evidence rather favours the derivation of the Anglesea boulders from dykes in that island." A hornblende picrite of a somewhat similar character occurs also in situ at Penarfynydd, on the south-west coast of Carnarvonshire. So it is probable that the boulder found near St David's has been carried by the agency of ice from Carnarvonshire or Anglesea. THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 79 The writer discovered a big boulder of picrite on Strumble Head, on a piece of uncultivated ground a little north of Tre-sinwen farm, by the side of the pathway leading to the coastguard station at the Head. The boulder had been broken, and now lies in several pieces — -the biggest piece measuring roughly 3|- x 3 x 2 feet — and in its original condition it must have been much bigger. Another boulder of picrite, very similar in appearance and size, and also broken, was discovered on a field lying a little to the north-west of Tre-seissyllt, between the farm and the coast north of Aber- bach. A microscopic section of this rock revealed the presence of the following minerals : — brown hornblende, strongly dichroic ; augite, nearly colourless ; olivine, not very abundant, and always very much serpentinised and of a greenish colour ; magnetite ; a chloritic mineral, which is evidently an alteration product, and a little plagioclase felspar. The chloritic mineral is greenish in colour and markedly dichroic. The general appearance of the rock as seen in section was very different to that of the St David's picrite : the augite does not show such perfect forms in the former as in the latter, and there is rather more olivine in the St David's rock. The Strumble Head picrite boulder is rather more like some of the Penarfynydd speci- mens as seen in section, but there is not so much olivine in the former, and the pcecilitic structure which often characterises the latter is not seen in the former. But it is highly probable that the Strumble Head picrites have also been borne from Lleyn or Anglesea. The two specimens found on Strumble Head lie about three miles apart, in a line whose direction is north-north-east to south-south-west. Near the Tre-seissyllt boulder of picrite lay a boulder of olivine-gabbro, also broken to pieces by blasting. The newly-exposed faces were remarkably striking, and the crystals are very fresh. The rock is quite unlike the gabbros found in Pembroke- shire. A microscopic section showed beautifully fresh olivine crystals, and the rock is undoubtedly of Tertiary age, and has probably come either from the Western Isles of Scotland or from the north-east of Ireland. 2. Erratics in the Drift. — As might be expected, the majority of the boulders found in the drift deposits are of local origin. They occur abundantly in the Upper Boulder-Clay and Rubbly-Drift, and in the sands and gravels, and to a less extent in the Lower Boulder-Clay. The grits, shales, and slaty rocks of Pembrokeshire are very similar in appearance to rocks of a like nature from North Wales, and the same is true of some of the igneous rocks, especially the diabases and some of the lavas. It is thus quite possible that among the boulders found imbedded in the drift many North Wales rocks may be represented, though there is no means of distinguishing them readily from the boulders of local origin. This was suggested to the writer by the discovery of boulders of a diabase rock in the boulder-clay exposed at Cardigan. To the naked eye this diabase seemed very like that found to the south-west in Pembrokeshire. But no such rock is known to occur anywhere nearer Cardigan than Newport — nine or ten miles to the south-west — and so these boulders must have come from the north. This is made all the more probable by the discovery of boulders of 80 DR T. J. JEHU ON what are undoubtedly northern rocks associated with these boulders of diabase at Cardigan. Of the erratics the most important discovery in the drift was that of a small boulder of the Ailsa Craig riebeckite rock, or paisanite. This was found in the bluish boulder- clay, near the surface, at Clun-Bach moor, St Nicholas, near the south-western end of Strumble Head. The specimen was sliced, and as seen under the microscope it is identical with specimens obtained from Ailsa Craig. Boulders of hornblende-porphyrites from the south-west of Scotland occur in the Lower Boulder-Clay as far east as Cardigan, and are found even oftener in the Upper stony Boulder-Clay. Several varieties are seen, all of which can be matched in Wig- townshire and Kirkcudbrightshire. But the erratics which are most commonly met with in the drift are reddish granophyres, quartz-porphyries and micro-granites. Manv of these are dyke-rocks, and it is very difficult to trace them to their source. Some have certainly come from Ireland, and some most probably from the south-west and west of Scotland. Lake District rocks and North Wales rocks are not so well represented. In the clay-pits of Bhos Isaf, near Dinas, excavated in the Lower Boulder-Clay, a reddish granophyre was found, which, under the microscope, resembles very much some of the granophyres of Mull. And in the Pen Creigiau gravel-pit boulders of granophyre occur, which have come from the Carlingford district, Ireland. One was sliced, and the microscopic characters seen were identical with that of the granophyre of Barnavaine, Carlingford. These reddish granophyres are found in the drift throughout the area, but seem to be rather scarcer in the extreme west. As the writer had not much opportunity of comparing them with Irish rocks, he sent them to Prof. Watts, of Birmingham University, who very kindly examined them. Many of the granophyres, he thinks, can be matched in the Carlingford mountains ; others bear more resemblance to the Tertiary granophyres of the Inner Hebrides. The reddish quartz-porphyries appeared to him to be like the varieties of the quartz -porphyry of Cushendale in Antrim ; and among the boulders of micro-granite he noted two which are likely to have come from the mass of micro-granite at Cushendun in Antrim. He is of opinion also that the Old Red conglomerate of Cushendun might, when broken up, present examples of many of the types of boulders found in the drift of Pembrokeshire. A pebble of muscovite-granite, probably the Foxdale granite of the Isle of Man, was obtained from the gravel-pit at Pen Creigiau, over 600 feet above sea-level. Examples of Millstone Grit were obtained here also, and in the Cardigan clay-pit boulders of Carboniferous Limestone are common, often with the fossils well preserved. On the beach at Gwbert, not far from Cardigan, boulders and pebbles of Carboniferous Limestone are common. These must have come from Ireland or from carboniferous rocks which are hidden at the bottom of the Irish Sea. It is hardly likely that they have come from the small exposures bordering the Menai Straits in North Wales. Chalk-flints occur everywhere — in both boulder-clays, in the sands and gravels, and THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 81 on the beaches. These also must be derived from the north-east of Ireland and from rocks hidden under the Irish Sea. From the cliff of Upper Boulder-Clay at Porth-lisky a boulder of olivine-dolerite was obtained, microscopic sections of which show very fresh olivine. It is certainly a Tertiary rock, and has probably come either from the north-east of Ireland or from the western isles of Scotland. 3. Erratics found on the shores along the coast. — These are especially abundant at those places where cliffs of boulder-clay or drift are seen. Many of them are found lying just at the foot of the cliffs, having only recently fallen from them ; and others which were picked up as pebbles on the beach have doubtless, for most part, been derived from the drift also. Ailsa Craig, Riebeckite Rock or Paisanite . . . found on AbermaAvr beach (frequent), Aberfelin beach, Porth-y-Rhaw beach. (It is interesting to note that a boulder of this was also found in the Lower Boulder- Clay at St Nicholas, not far from Abermawr.) Granites from the Dalbeattie area, several varieties . found at Pwll Gwaelod beach (frequent), Aber- bach beach (near Dinas). Granites of Galloway type . . . ... . ,, Whitesand Bay, Aberbach (near Dinas), Pwll Gwaelod, Gwbert (near Cardi- gan). ,, Abermawr. A fine specimen of a Mica-hornblende-Granite, identical with that of Auchencairn, Kirkcudbrightshire, Mull of Galloway Granite ...... Another variety from same area ..... A Gneissose Granite from Criffel ..... Granite or Quartz-Diorite from head of Loch Doon, South of Scotland, Biotite Granite, Loch Dee, South of Scotland A Diorite identical with that of a dyke near Gutchen Isle, Colvend shore, south of Dalbeatie, Other Diorites from the Galloway area .... Hornblende-porphyrite identical with one found south of Castle Douglas, Kirkcudbrightshire, Other Hornblende-porphyrites of the Galloway country . Hornblende-biotite-porphyrite, "Wigtownshire . Silurian grits, South-West of Scotland .... Muscovite-granite, Foxdale, Isle of Man Andesites, Rhyolites, and altered Tuffs of the Borrowdale series, Reddish Quartz-porphyry, probably from Cushendale, Antrim, Reddish granophyres and micro-granites, mostly North- East Ireland, but some possibly from West of Scotland, A gneissose Grit — locality unknown .... Carboniferous Limestone ...... Gannister ......... A Muscovite-granite, with microcline and some biotite — locality unknown. Pwll Gwaelod. Gwbert (near Cardigan). Pwll Gwaelod. Pwll Gwaelod. Abermawr, Gwbert (near Cardigan). Abermawr, Whitesand Bay. Abermawr (frequent), Aberbach (near Dinas), Abereiddy. Pwll Lan-ddu. Pwll Gwaelod, Aberbach (near Dinas), Abermawr (frequent), Whitesand Bay. Pwll Gwaelod. Abermawr. Gwbert (near Cardigan). Abermawr, Aberbach (near Dinas). Porth-y-Rhaw, Abermawr, Whitesand Bay. Pwll Lan-ddu, Gwbert (near Cardigan), Aberbach (near Dinas). Porth Sele Gwbert (near Cardigan). Abermawr. Abermawr. 82 DR T. J. JEHU ON The most striking fact in connection with the erratics is that so many of them can be traced to the south-west of Scotland. The Ailsa Craig paisanite has been obtained in the boulder-clay, and is frequently met with on some of the beaches, especially at Abermawr. The granites, diorites, and porphyrites of the Galloway country are also well rejDresented, boulders being found which represent the three principal massifs, namely, (l) Dalbeattie and CrifFel, (2) Cairns Muir of Fleet and New Galloway, and (3) Loch Doon and Loch Dee, and in addition some from smaller exposures, such as that of the Mull of Galloway. The other region from which the boulders have travelled is the north-east of Ireland, and its rocks are represented in Pembrokeshire by reddish granophyres, quartz- porphyries, and micro-granites. A few boulders are found also which have almost certainly come ultimately from the Western Isles of Scotland. It is a noticeable feature that the Lake District rocks are but poorly represented, and the same is apparently true of those of North Wales. Many of the boulders and pebbles, such as those of Carboniferous Limestone and the chalk-flints, may have been torn up from the bed of the Irish Sea. VI. General Conclusions. The facts adduced in this paper prove conclusively that northern Pembroke- shire has been the theatre of glacial action to an extent greater than had previously been supposed. Glacial deposits cover the ground in that part much in the same way as they do further north, and present very similar characteristics. Here also we meet with a tripartite division of the deposits, namely, a Lower Boulder-Clay, Inter- mediate Sands and Gravels, and an Upper Boulder-Clay and Rubble-Drift, reminding us of the tripartite division found at so many places further north on both the east and west sides of England, and in North Wales. But in the present state of our knowledge it is very difficult to correlate the deposits found in one area with those found in another area ; and it is not safe to assume that the Sands and Gravels always represent any definite horizon in the glacial series. Of the deposits which have been described it is the Lower Boulder-Clay which has the widest extension ; it covers much of the lower grounds inland, and is often hidden under the other accumulations. It follows the slope of the ground, and a little below Pen Creigiau Cemmaes it attains an elevation of nearly 600 feet above sea-level. The series of sands and gravels is a very variable one. Often they taper or die away suddenly into a stony or loamy drift, and at places are absent altogether. They attain their greatest elevation near the east end of the area at Pen Creigiau, where the sands reach a level of over 640 feet, and are followed immediately below by coarse shingly gravel. The Upper Boulder-Clay, where the sands and gravels are absent, is some- times seen to rest immediately upon and coalesce with the Lower Boulder-Clay, so that THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 83 it becomes somewhat difficult to separate them. But the distribution of the Upper Drift is not so wide ; it is only met with here and there, and the true Upper Boulder-Clay is often replaced laterally by Bubbly-Drift. The more sporadic occurrence of the Upper Boulder- Clay is probably due in part to the fact that it has suffered more from denudation. The Lower Boulder-Clay is undoubtedly the product of an ice-sheet, and it has all the characteristics of a true ground-moraine. It is remarkably tough and homo- geneous, and shows no traces of stratification, and it has all the appearance of having undergone great compression. The included stones are often intensely glaciated, and are sub-angular rather than rounded in form. The fact that fragments of marine shells occur in the clay proves that the ice which gave rise to it must have travelled over a sea-bottom. On the other hand, the bits of woody matter sometimes seen embedded in the tough clay to a depth of 18 or 20 feet suggest that vegetation grew on the land bordering the sea, before the advent of the ice or during an interglacial period, and that some fragments of this found their way, by means of streams or otherwise, to the sea-bottom, where they lay in the path of the ice. Or they may have been derived from the remains of a submerged forest. The included erratic stones help us to follow the direction from which the ice came, and the occurrence of boulders from the south-west of Scotland and from the north-east of Ireland in the Lower Boulder-Clay and Drift as far east as Cardigan, and the discovery of fragments of marine shells in the Lower Boulder-Clay exposed at the brickworks near that town, make it clear that the whole of northern Pembrokeshire was buried underneath an ice-sheet coming from the north. The view held by Carvill Lewis, that the Irish Sea glacier (as he termed it) extended no further south than the extremity of Lleyn in Carnarvonshire, is shown to be inaccurate. And though Professor James G-eikie makes the mer de glace which overwhelmed Anglesea flow down St George's Channel, to a limit reaching beyond the south-west of Wales, he only indicates it as crossing the extreme west of Pembrokeshire at St David's Head. But the facts just mentioned show that this mer de glace must have passed over a great deal more of Pembrokeshire than St David's Head. It invaded northern Pembrokeshire along its whole extent, and even encroached on Cardiganshire to the east, and its trail is evident in the tough dark-blue homogeneous boulder-clay, with its northern erratics and the broken shells derived from the sea-bottom over which the ice travelled. How much further south this typical boulder-clay or ground-moraine extends is a point which must be left to future investigation. This mer de glace was of course the southward extension of that ice- sheet which filled the northern basin of the Irish Sea, and which has been described by Professor James Geikie and other workers in Glacial Geology. The latest results published are those of the investigations of Mr Lamplugh in the Isle of Man, and these have appeared in his Survey Memoir on the Geology of that Island. His observations on the Irish Sea Glacier are of great interest and importance, and throw light even on 84 DR T. J. JEHU ON what occurred to the south of his area. Speaking of the conditions which obtained in the northern part of the Irish Sea at the beginning of the Glacial period he says, " Along the shores an ice-foot probably formed in the winter and broke away in the summer into floes, which distributed their burden of rock-fragments broadcast over the sea-floor. This seems to be the explanation of the universally wide dispersal of the fragments from Ailsa Craig, which have been recognised in the drift almost all round the northern part of the Irish Sea basin, in Ireland and Wales, as well as in the Isle of Man. The sea-girt precipices of splintering rock in Ailsa would not fail to cast off a load upon an ice-foot below ; and thus these fragments became strewn over the sea-floor almost as widely as the shells, and were subsequently carried by the ice-sheet into nearly every district to the southward where the shells were carried " (p. 370). This helps to explain also in a satisfactory way the occurrence of fragments from Ailsa Craig and of boulders from the north-east of Ireland, from the south-west of Scotland, and even possibly from the Inner Hebrides, in the drift, and on the beaches of northern Pembrokeshire. For the ice-sheet as it advanced would pick up any such fragments which had been previously strewn over the sea-bottom by ice-floes, and would carry them southwards on to the land as it carried the shell-fragments. Lamplugh estimates that in the neighbourhood of the Isle of Man the ice -sheet, at its maximum, must have attained an elevation of not less than 2000 to 3000 feet above the present level of the sea, and the general direction of the ice- movement was from north-north-west to south-south-east. He points out that " the West-British Ice-sheet probably attained its ultimate dimensions mainly from the accretion of snowfall upon its surface, and in only a minor degree from the inflow of tributary glaciers." He calls attention to certain results which help us to understand the south- ward extension of the ice-sheet far enough to overwhelm Pembrokeshire. " As the British ice-sheets must always have received their increment principally from the moist Atlantic winds, it seems probable that, without any change of climate, the centre of greatest accumulation, and consequently of maximum glaciation, would tend to shift steadily westward and south-westward as the icy plateau rose higher in the path of the moisture-laden winds and compelled their earlier precipitation. This effect would, moreover, be accentuated by the obliteration of the open water in the sea-basins to the eastward. The West-British sheet might from this cause go on increasing, while its East-British and Pennine equivalents were already diminishing from lack of sufficient snowfall. . . . For the above reason, the shrinkage of the ice-sheet covering the Isle of Man is likely to have commenced while the Welsh and Ivernian sheets were still increasing." Although the ice in the southern part of the Irish Sea basin did not probably attain such a great thickness as the ice in the northern area did, nevertheless all the evidence goes to show that even as far south as Pembrokeshire it must have reached a considerable elevation. The presence of drift material at Pen Creigiau, at an altitude of over 600 feet above sea-level, indicates that the ice-sheet here was in all probability not much less than 1000 feet in thick- THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 85 ness, even if we allow that the land at that time stood at a somewhat lower level than it does at the present day. Mr Lamplugh gives a sketch-map of the Irish Sea (as far south as the Lleyn promontory), showing glacial strise and probable direction of the ice-movement. The ice which streamed over the Isle of Man from the north is shown as usual to have travelled south and to have overwhelmed Anglesea, being here diverted so as to move more to the south-west on account of the opposition of the ice coming down to meet it from the mountains of Snowdonia. On the western side of the Irish Sea basin the strise indicate that the ice moved from the land on the eastern seaboard of Ireland, and took a course from north-north-west to south- south-east, and coalesced with that which passed over Anglesea and Lleyn. Its course southwards from this limit is not shown. But in Professor Geikie's map the Irish ice is made to bend back to the south as a result of its meeting with that part of the ice-sheet which flowed over Anglesea, and the northern ice is shown as passing down to the west of Cardigan Bay, on account of the presence of the ice flowing west from Merionethshire and Central Wales. But the investigations carried out on the glaciation of Pembrokeshire make it clear that the Irish ice was not bent back so sharply, but, on the contrary, it continued in its original course from north-north-west to south-south-east, whilst the ice from the north was forced to invade Cardigan Bay, and must therefore have shouldered in the Welsh ice again upon the mainland. This is proved by the direction of the striae seen along the coast, as well as by the presence of boulders of igneous rock from Ireland and the south of Scotland in the drift as far east as Cardigan. Again the presence of chalk-flint throughout the area is evidence in the same direction, for these must have come from the north-east of Ireland or from the bed of the Irish Sea ; and it is possibly from this bed that the boulders of Carboniferous Limestone which are seen so abundantly at Cardigan have come. In this connection it is interesting to recall the presence of fragments of Millstone Grit in the gravels at Pen Creigiau. Our knowledge of the glaciation of Ireland is as yet very imperfect, and it is difficult to estimate what volume of ice passed seawards from its eastern border. At the present day the rainfall over Ireland is very excessive, and so it seems probable that the snowfall was likewise excessive during glacial times. This would give rise to a proportionately large ice-sheet moving outwards in all directions, and so it is quite possible that the amount of ice which found its way into the Irish Sea basin was considerably greater than has been generally supposed. And in the southern part of the basin it would to some extent oppose the passage of the western ice which overflowed Anglesea and the end of Lleyn in a south-westerly direction, and cause it to turn a little more to the south so as to travel over Cardigan Bay. The confluent sheet, forming by the junction of the Northern ice, the Irish ice, and to some extent the Welsh ice, would invade northern Pembrokeshire in the direction which is shown by the striae along the coast near St David's, namely, from north-west to south-east, or perhaps from north-north-west to south-south-east. This would explain the transport of boulders from the St David's Head gabbro south- TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 4). 14 S6 DE T. J. JEHU ON eastwards to the neighbourhood of Caer-bwdi Bay. In this connection it may be mentioned that Mr J. Harris, in a report on erratics in South Wales, which appeared in the British Association Reports, 1898, refers to some boulders found at Pencoed, near Bridge-end, Glamorganshire. Microscopic sections of some of these were prepared and sent to petrologists for examination. The result of this is given as follows : — ' " One was identified with the gabbro of St David's Head ; a felsite bore some resemblance to the pre-Cambrian rocks of Pembrokeshire ; two or three acid rocks, brecciated felsites, and tuffs are very like those of the Lleyn promontory." From these data it is concluded that the transport of boulders was from the west or north-west. If one of the boulders found near Bridge-end is accurately identified as belonging to the St David's Head gabbro, it is a most remarkable fact. It is hardly safe to draw any conclusion until it has some further corroboration. The more or less loose materials covering the bottom of the sea, which existed before the advance of the ice, would become incorporated into the lower layers of the ice-sheet. And as the ice was very thick, and moved onward slowly, it would exert a great pressure over its bed, with the result that much of the rocky floor would be torn away, and much of the material ground up and pulverised to form the typical ground moraine. The shell - banks which occurred on the sea - bottom would be destroyed, and the marine detritus would be carried forward under the ice or in the ice. This accounts for the presence of shell-fragments at places in the Lower Boulder- Clay. And the most natural explanation of the shelly sands and gravels is that they represent the material of a sea-bottom, carried onwards and upwards to their present position by an ice-sheet, and re-arranged by fluvio-glacial action. That is to say, they are remanies derived from the bottom-moraine of an ice-sheet which had travelled over a sea-floor. Similar sands and gravels have been found at many other places on the west side of the island, and they have given rise to much discussion — notably those found at Moel Tryfan in Carnarvonshire. The Pem- brokeshire series differ from those of Moel Tryfan in that they are found overlying the well-marked stiff Lower Boulder-Clay. The most remarkable feature in connection with the Moel Tryfan beds is the great elevation at which they are found — 1350 feet above sea-level. In Pembrokeshire the greatest height at which they have been met with is 642 feet at Pen Creigiau, four miles south-west of Cardigan. The mode of origin of such sands and gravels has been one of the most vexed questions in Glacial Geology. Some writers, such as Mackintosh, T. M. Reade, and others, have argued that the sands and gravels represent marine deposits laid down in place during a great submergence. It is admitted, even by the opponents of that theory, that a partial submergence took place during Glacial times, but to what extent is not known, and there is no evidence to show that it meant a sinking of the land in Carnarvonshire to as much as 1350 feet below its present level. And the partial subsidence which is allowed is generally thought to have diminished towards the south. In Pembrokeshire no evidence can be seen along the coast which would lead us to THE GLACIAL DEPOSITS OF NORTHERN PEMBROKESHIRE. 87 believe that there had been a subsidence of 640 feet below the present level, so as to account for the marine deposition of the beds found at Pen Creigiau. Further, if the advocates of a submergence point to the presence of marine shells at places in the sands and gravels as a proof, on the ground that the shells must have been altogether destroyed if carried beneath the ice in the morainic debris, one asks then how they can account for the presence of fragments of shells — some large enough to be identified — in the tough blue boulder- clay underlying the sands and gravels ? No one who looks upon that clay, as exposed, for instance, at the Cardigan brickworks, can doubt for a moment that it is the product of an ice-sheet. Not a trace of stratification can be seen in it, nor is there any character which suggests even the possibility of its being the result of marine deposition. And yet marine shells are seen imbedded in the clay. This, as has been already pointed out, is due to the fact that the ice-sheet, of which this clay is the bottom-moraine, travelled over a pre-existing sea-bottom. The fact that everywhere the shells are very broken and much rolled is hardly compatible with the view that they are now found in or near the positions in which the molluscs themselves lived. And it is worthy of note that the Lamellibranch shells obtained in these sands and gravels are never found with the two valves in apposition, as one might expect to find if they lie in ordinary sea deposits. Again it has often been pointed out that it is a significant fact that deposits of this kind only occur in glaciated areas, and that wher- ever broken shells are found, with them there also we find far-travelled erratics present. And this is to a marked extent the case in Pembrokeshire. Mr J. F. Blake (Geol. Mag., vol. x., 1893, p. 267) concluded that the shelly sand at Moel Tryfan had been pushed up in front of the advancing glacier, and that, as a result of this glacier meeting that which came out of the Bettws Garmon valley, the sand got pushed into a protected corner and was left there. But in northern Pembrokeshire the sands and gravels are found scattered in patches over a wide area, and are frequently well bedded. Here they are the products of the washing and re-sorting of infra- and intra-glacial detritus. This may have gone on partly under the ice, but it would no doubt take place to a great extent at the time of the melting of the ice-sheet, when large streams would issue from the margins of the glacier and re-arrange much of the superficial deposits left on the surface of the land. The Upper Boulder-Clay is so sporadic in its occurrence that it is difficult to draw any definite conclusions with regard to it. It may possibly represent a second advance of the ice-sheet after an interval of less severe glacial condibions. It is far more stony than the Lower Boulder-Clay, and in places passes into Rubbly -Drift. This Rubbly-Drift is very similar to that found by Lamplugh in the Isle of Man, and is probably " the remanie deposit of the ice-sheet modified by sub-aerial agencies." At the time of the final disappearance of the ice, torrential waters must have overflown parts of the surface, and the rubble is probably to be attributed in part to the action of these waters. Morainic material would become mixed up with rock debris, formed by ordinary weathering processes, and the whole mass would be re-arranged, and in places sifted by the waters. Trans. Roy. Soc. Edin. Vol. XLI. 2 o o 2- S3 3 B H •-3 1-5 H C n i »1 1 * > ' 4 :»/•'> » •> 03 r;sA i " -; P3 ( 89 ) V. — Spectroscopic Observations of the Rotation of the Sun. By Dr J. Halm, Assistant Astronomer at the Eoyal Observatory, and Lecturer in Astronomy at the University, Edinburgh. Communicated by The Astronomer Royal for Scotland. (MS. received February 19, 1904. Read March 21, 1904. Issued separately May 4, 1904.) The causes of the peculiarities of the solar rotation exhibited by the superficial layers of the sun's body must still be considered unknown, notwithstanding recent interesting attempts at explanation. This is no doubt partly due to the difficulties of the hydro- dynamical problem placed before the mathematician in an investigation of such complexity as the movements of particles in a rotating fluid subject to energetic convection. But it must also be conceded that the observational data available for a basis of mathematical, or even speculative, research are still so scanty, that for this reason alone we may perhaps not feel surprised at the failure so far of theoretical attempts. The first empirical demonstration of the peculiar law which apparently governs the rotation of our luminary was given by Carrington, whose important investigations were taken up and extended by Spoerer. Their observations brought to light the main character of the peculiarity of solar rotation, viz., the decrease of the angular velocity from the equator towards the poles. An objection has, however, been raised against the value of their results with regard to the general rotation of the sun's photosphere, on the ground that they were obtained from the observed movements of solar spots. It was justly urged that spots have " proper motions " which preclude their adoption as points of reference. Besides, we must remember that these spots were visible only within an equatorial zone of about ±50° or 40° latitude, and that therefore the polar regions remained inaccessible by this method. This difficulty and limitation was over- come by the ingenious application of the spectroscope to the problem, which we owe to Professor Duner. His conclusions are based on the displacements shown by the Fraunhofer lines at the solar limb, where the gases producing these absorptions are carried by the rotation either towards us or from us. His results are, it is true, not directly comparable to those derived from the movements of the spots, because both refer most probably to different levels, and therefore perhaps to different conditions of motion ; but the great advantage of the spectroscopic method seems to me to lie in the fact that we always measure at the same level, wherever this level may be — a point on which we are by no means certain in the case of the spots. Besides, we are independent of the uncontrollable vicissitudes of proper motions, and we are able to extend the investiga- tions from the equator to the immediate vicinity of the poles. The results obtained by Professor Duner may be summarised by the statement that the retardation of the angular velocity discovered by Carrington and Spoerer was found to be also shared by the photospheric layer emitting the Fraunhofer lines, and that the amount of this TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 5). 15 90 DR J. HALM ON " lag" appeared to increase continually towards the poles. So far, then, as the general character of this peculiarity goes, the question appears to be empirically settled. But there remains still another, and, as I think, not less important question. Are we allowed to suppose that the surface rotation of the sun remains unaffected by the periodic changes of solar activity ? Judging the question from a purely logical point of view, we are almost bound to answer it in the negative. It seems to me difficult to imaoine that such violent disturbances of the normal conditions of convection as we perceive in solar eruptions and spots, and the consequent displacements of matter in the solar olobe, should have no influence on the distribution of the rotational velocities at the surface. A careful study of the behaviour of the solar rotation during a cycle of activity may probably teach us far more about the causes and the seat of these solar Spectrum of receding limb. Solar limbs. Spectrum of approach- ing limb. Fig. 1. Group of lines as seen in the viewing telescope. (1 and 3 are telluric, 2 and 4 solar line s. ) disturbances than the whole array of statistical facts regarding the periodic displays of dynamical phenomena at the surface which are now in our possession. Professor Duner's observations, covering a period of three years, during which next to no change took place in the activity of the sun, cannot give an answer to this question. I there- fore thought it a promising venture, the success of which seemed to me in some way guaranteed by the great accuracy and consistency of Duner's results, to extend these observations over a time of more pronounced changes of solar activity. The following contains a description of the results so far obtained, and of the instrument employed in my investigations. The observations were begun in August 1901, and so far carried on to the end of 1903. Although this interval of time is not greater than that covered by Duner's work, there is this essential difference, that the year 1903 was characterised by an abrupt and violent increase of solar disturbances after a pronounced and persistent calm during 1901-2. This fact, as we shall see, has an important bearing on the results, which SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 91 seem to me so remarkable and unexpected, that I trust this early publication may induce other observatories to take part in the work, at this opportune time of the beginning of what seems to be an energetic sunspot cycle. As I intended to make the following observations directly comparable to those of Professor Duner, I have used the same group of lines. The wave-lengths of these are given on page 55 of his treatise, " Eecherches sur la rotation du Soleil." # The measurements therefore refer to the same photospheric level. Fig. 1 represents this group of lines as seen in the viewing telescope. The displacements of the solar lines shown in the figure correspond approximately to the shift at the solar equator. With regard to the instrument, some deviations from Duner's arrangement have been suggested by the apparatus at my disposal. Since they increase the stability of the instrumental plant, these alterations may be considered as essential improvements. The most important of them was attained by the use of a siderostat. Duner's spectro- scope was mounted on the great refractor of the Lund Observatory. The extraordinary dimensions of the apparatus gave rise to flexures which were bound to lessen the accuracy of the observations. The influence of such flexures is, of course, avoided if the whole spectral apparatus, including the front telescope, is mounted on fixed and insulated tables, and if the solar light is thrown upon the object-glass of the telescope by means of a siderostat. The considerable advantage of such an arrangement is indeed shown by the fact that the probable error of a single observation appears to be only half the probable error of one of Professor Duner's measurements. This increase of accuracy has to be ascribed chiefly to the greater stability of the instrument, and perhaps also to the easy and comfortable position of the observer during the observations. A second essential departure from the design of the apparatus used by Professor Duner is to be found in the arrangement by which the focal images of the two opposite limbs of the sun are thrown upon the slit of the spectroscope. Duner employed a system of right-angled prisms arranged according to a device previously suggested by Langley. By successive inner reflections from the hypotenuse surfaces of these prisms the light of the solar limbs can be thrown upon neighbouring points near the centre of the slit, which lies in the optical axis of the telescope. In a much simpler way, however, this same purpose can be attained by using as a front telescope a heliometer of sufficient optical power. By separating the halves of the object-glass, we can at once bring opposite points of the solar limb into contact, and these images may be thrown upon the centre of the slit without any further auxiliary apparatus. Besides, by altering the position angle of the heliometer, all the opposite points of the solar disc can be successively brought into contact. Thus we are enabled to determine the rotational velocity for any desired heliographic latitude simply by turning the heliometer into a position which corresponds to that latitude. It seemed advantageous to throw the two solar images at each observation into such a position that the line joining their centres coincided with the slit. This could be * Nova Acta Eegiae Societatis Scientiarum Upsalie?isis, 3rd series, vol. xiv. fasc. ii., 1891. 92 DR J- HALM ON effected by bringing a large-sized, right-angled prism into the cone of light between the object-glass of the heliometer and the focus. By turning the prism, the solar images projected upon the slit-plate could be brought into the desired position, which is represented in the accompanying fig. 2. I may say that in order to attain sufficient accuracy of this adjustment, two lines parallel to the slit and at equal distances from it had been engraved upon the slit-plate. The solar images were brought into such a position that the two lines cut off equal segments. This arrangement was quite sufficient to guarantee the heliographic latitudes of the points measured within a fraction of a degree. Such small errors in the adjustment have, however, no appreciable effect on the observations, because they displace the measured point on the one limb exactly as much towards the equator as they displace the point measured on the other limb towards the pole. The total displacement therefore still agrees practically with that we should have obtained if no such error had been present. This consideration, however, does not apply to points exactly on the equator. Fig. 2. The heliometer employed in these observations has been kindly lent to this Observatory by the Hon. Lord M'Laren, Judge of the High Court of Session of Scotland. It is the instrument used by Sir David Gill in the Mauritius Expedition, 1874, and a full description of it may be found in vol. ii. of the Dunecht Observatory publications. Its aperture is 4*2 inches, and the focal length about 64 inches. The optical quality of the glass is exceedingly good. For the purpose of the observations the eye-end part of the heliometer had to be removed, so that the focus came to fall outside the main tube. The instrument is mounted upon a cast-iron table, adjustable in two horizontal directions at right angles, with the optical axis parallel to the line of the meridian, the object-glass towards the north. As a collimator, a telescope is used of 4 inches aperture and 50'5 inches focal length. The eye- end tube, which carries the slit, is adjustable by means of a focussing screw. The cylinder of rays, emanating from the object-glass of the collima- tor, is thrown upon a Rowland plane grating of speculum metal, 5 inches long, 3^ inches broad, with 14,438 lines to the inch. The surface of the grating is perpendicular to the axis of the collimator. I have used the spectrum of the third order, which, upon the one side of this grating, is remarkably bright and well-defined. The disturbing effect of the overlapping violet spectrum of the higher order was found to be sufficiently elim- SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 93 inated by a plane glass of purple colour fixed in front of the eye-piece. The grating is mounted upon a stand, turning on a vertical axis in such a way that the axis of rotation coincides exactly with the plane of the grating. This stand carries a horizontal circle, divided from 5 to 5 minutes, which can be read by two diametrically opposite microscopes. By this arrangement a differential determination of wave-lengths is made possible. The viewing telescope, which is horizontally mounted on another isolated stand, has an object-glass of 4*1 inches aperture and 60 inches focal length. A Cooke wire microm- eter is used for measuring the distances between the lines. After various preliminary trials with crossed and parallel wires, the pointing on the spectral lines by means of sufficiently close parallel wires seemed to me to yield the best results. The observation consists in bringing, by turning the micrometer screw, the centre of the space between two close wires exactly upon the line to be measured, in the same way as the meridian circle observer sets the division of the circle between the wires of the microscope. Care has to be taken, however, that the setting is made exactly on the solar limb. I have found that this may be done without the least difficulty. The errors of the micrometer screw have been repeatedly determined, with the result that the progressive error may be considered as negligible, but that there are indications of a small periodic error. In order to eliminate the effect of the latter directly from the observations, I have invariably observed the lines with two pairs of wires which were at a distance of exactly 1 1- turns of the screw. As is well known, the periodic error is practically eliminated from the arithmetical mean of the observations made with two sets of wires thus arranged. The value of a revolution of the screw, expressed in wave-lengths, has been determined by measurements of the distances of neighbouring spectral lines, by a method which is fully described in Professor Duner's paper. Such measurements will be continued in the future, but there can be no doubt already that the provisional value here adopted must be sufficiently correct, and that therefore the computed values of the velocities may not have to be altered in the final discussion. The value of the screw is subject to small variations, chiefly owing to temperature fluctuations. These have, however, been taken into account in the reductions in the following way : — Each set of observations yields a very accurate determination of the normal distances between the four lines of the group measured. Since the wave-lengths of these lines are known, the value of a revolution of the screw expressed in wave-lengths may be found by dividing the measured distances of each pair of lines into the difference of their wave-lengths. Thus values of a turn of the screw are found for different points within the group, and from these the value to be applied in the reduction can be easily derived. The computation of the position angles of the various points measured along the solar limb requires the knowledge of the rotation of the field of the siderostat at the time of the observation. Obviously, what is required is the diurnal rotation of the hour circle of the sun, and therefore also of the pole of the heavens, round the centre of the field. The data required for this computation have been supplied by Cornu in his 94 DR J. HALM ON paper on the " Law of Diurnal Rotation of the Optical Field of the Siderostat and Heliostat" {AstrophysicalJournal, vol. xi., 1900, pp. 148-162). It appears from his calculations that in our case, where a siderostat oriented in the meridian has been used, the angle Y at the centre of the field between the reflected image of the meridian and that of the hour circle of the sun is expressed by the formula, tan-£Y = K tan|£, where t is the hour angle, reckoned positive towards the west, and K _ Bin}(0-j?) sin H0+P)' </> being the latitude of the place and p being the polar distance of the sun. These equations determine the amount and the direction of the rotation. We are thereby enabled to fix the position of the north point of the solar disc as it appears in the field of the siderostat for any time of the day. The values Y can be tabulated with the arguments t and declination of the sun. The position circle of the heliometer is so oriented that the diameter 0°-180° falls in the plane of the meridian. The angular distance of the point under observation from the north point of the solar disc, that is, the position angle P of the point, is then found by the expression P=90°-n+Y, where II is the reading of the heliometer position circle. The value of P being found, the heliographic latitude of the point observed can be directly computed from the formulae given by Professor Duner in his treatise, where he also exhibits extensive tables, greatly facilitating these computations. I need not, therefore, enter more fully upon this part of the reductions. I shall now briefly describe the way in which observations were performed with this instrument. The sunlight reflected from the siderostat mounted on the main platform of the Observatory is thrown into a meridional direction upon a window of 6 inches aperture in the north wall of the great optical room, and falls upon the object-glass of the heliometer placed immediately behind this window. The halves of the object-glass are screwed apart until the two solar images are nearly in contact, but still separated by a narrow space, which in the viewing telescope appears as a dark horizontal band between the spectra of the two limbs. The right-angled prism mentioned above is then turned and the heliometer adjusted so that the solar images fall upon the slit in the position indicated in fig. 2. After reading the position circle of the heliometer, the observer commences the measurements by pointing the first pair of the micrometer wires on the four lines of the upper of the two spectra seen in the viewing telescope. He begins, say, with the left-hand line of the group, and proceeds towards the right. He then measures with the same pair the lines of the lower of the two spectra in the direction from right to left. The observations are afterwards repeated with the second pair of wires ; this time, however, they are begun on the lower spectrum from left to right, and finally the upper spectrum is measured from right to left. This arrangement, though it may SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 95 seem perhaps somewhat pedantic, has certainly the advantage of referring the mean of the observations on each limb to one and the same moment of time. Besides, approach- ing a line alternately in opposite directions must decrease the amount of the personal error of the observer in his judgment of the bisection. After the completion of these measurements, the halves of the heliometer object-glass are reversed and the time noted. The observations are then repeated in the same order as before. A complete set there- fore consists of 32 pointings, 1G on each limb. When the set is finished, the heliometer is turned 10° in position angle, and the solar images are again brought into the position of fig. 2. The observation of a full set, including the necessary adjustments of the instrument, requires on the average from 10 to 15 minutes. The observations of Professor Duner refer to six equidistant points on the solar limb between the equator and 75° latitude. The selection of these fixed heliographic latitudes was perhaps necessitated by the arrangement of his instrument, which required a previous computation of the difference of declination between the north limb of the sun and the point to be observed. Since, however, a previous knowledge of the heliographic latitude is not required in our case, I have preferred to proceed from 10 to 10 degrees on the position circle of the heliometer, without considering at all the heliographic position of the points thrown upon the slit. The readings of the position circle, together with the time of the observation, are sufficient to evaluate afterwards the true heliographic latitude. In this way, a uniform and continuous distribution of observations over the whole quadrant, from the equator to the pole, may be secured, and we are, I think, in a better position to ascertain the character of the velocity -curve by this method than by the observations of six single points of the quadrant. After this general description of the instrument and the method of observation, I shall now turn to the results. The measurements were commenced on 13th August 1901 and extended to 6th November 1903. During this time 564 determinations of the rotational velocity were made. The values obtained, expressed in kilometres per second, were divided into two groups ; the first group comprising the measurements made during 1901-2, the second those of the year 1903. The reason for this division is that the period 1901-2 was characterised by a low and protracted minimum of solar activity, while in the early part of 1903 the commencement of a new solar cycle was vigorously manifested by the appearance of large spots. It was therefore to be expected that, if indeed solar activity has an influence on the rotation of the sun, such a division into groups as I have made would show this influence more clearly than any other. In each group the individual values were arranged according to their heliographic latitudes, and from the materials thus collected normal values were formed by adequate combina- tions of single values into arithmetic means. The figures thus obtained are exhibited in the following table. 96 DR J. HALM ON Table I. 1901-2. 1903. Heliographic Linear Velocity No. of Heliographic Linear Velocity No. of Latitude. per sec. Obs. Latitude. per sec. Obs. km. km. 1-6° 1-908 33 3-7° 1-898 16 5*2 1-894 30 9-1 1-883 15 8-3 1-871 30 155 1-831 16 12-8 1-802 30 21-1 1-753 15 18-0 1-720 30 27-7 1-631 16 23-8 1-594 30 34-0 1-512 16 30-6 1-488 30 40-3 1-365 15 38-7 1-265 30 47-1 1-201 16 47-3 1-061 30 54-7 1-014 16 55-8 0-840 30 62-3 0-797 15 65-1 0-560 30 69-2 0-564 16 755 0-307 30 76-0 0-408 15 83-0 0-187 14 The values of the linear velocities were now plotted upon squared paper as ordinates, with the heliographic latitudes as abscissae, and in each group a curve was drawn repre- senting, as closely as possible, the observed ordinates. In this graphical form the results are represented in the accompanying fig. 3. It is seen at first glance that the two groups differ materially from each other. They agree fairly well only at the equator, but in higher latitudes the observations of 1903 show considerably larger values than those of the two preceding years. There is not a single exception to this rule. The evidence of these curves leaves therefore no doubt as to the reality of this systematic difference, which is also apparent from a consideration of the probable error. An in- vestigation of the observations of 1903 shows the probable error of a single observation to be ±0*070 km., a value which appears to be very nearly constant for all helio- graphic latitudes. From this we find the probable error of a mean value for the first group ±0*013 km., and for the second group ±0*018 km., while the observed systematic differences between the groups attain the considerable value 0*16 km. in middle latitudes. Professor Duner has also computed the probable error of his observations. I find from his figures that of a single observation to be ±0*138 km., also nearly constant for all latitudes. Considering that his single observations comprise twelve to twenty-four pointings of each line, as compared with only eight in my observations, we may conclude that the latter show at least double the accuracy of those of Professor Duner. As already pointed out, this favourable result is in my opinion only due to the greater stability and the more comfortable management of the instrumental plant. The influence of a possible error in the position angles shown by the heliometer has SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 97 also been closely investigated. The careful mounting and orientation of the instruments, it is true, did not warrant the assumption that such an error was to be seriously appre- hended. Besides, the arrangement of the observations was such that an error of this kind, even though it existed, should practically disappear from the mean values. This GM^-O. \~AMT-VLd j-nxrhM/nxi/ Vnxi AxnnAxyrf rotaZu^ruxL vJuxuXajla oZ^vrouL 2. titti JajurxmA, AhjU Axmojr* 1901-02 a/rv<L 1^)03 ' 19 18 ^x^ * ^J>Q 1.T 1.6 \ 1.5 \ \ 14 s \ 1.3 \ \ \ 1.2 * » \ 1.1 1.0 \ \ 0-9 0.8 ' \ \ \ \ 07 \ \ 0.6 \ \ \ \ 0.5 \ 0.4 \ \ , N^. 0.3 \ \ \ \ 0.2 T \ \ \ 01 \ \ \ N. \ o.o S \ \\ ts:)° 10° 20° 3o* 4o° 5o c 60' 70" 80 90 may be seen from the following consideration. Let the error of the position angle be Ap, so that a point on the solar limb whose computed distance from the equator, measured in the direction of the position angle, is II, is actually at a heliographic latitude II + Ap. If, now, we pass to the opposite side of the pole to a point whose computed distance from the equator, again measured in the direction of the position angle, is 180° — IT, such a point would actually be in the heliographic latitude II — Ap. Hence TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 5). ] 6 98 DR J. HALM ON our first point would be nearer the pole and the second nearer the equator than is shown by computation, and the arithmetic mean of both must therefore agree with the actual latitude. If, then, we arrange, as has indeed been done in these observations, the measurements so that equal numbers of points are observed on opposite sides of the poles, the mean values of the heliographic latitudes are practically uninfluenced by a systematic error of the position ang]e. The question, however, whether such an error actually exists, is readily answered by the observations. We have only to separate the QJ-UX-^t Micywruj/ Mvo uada) V Ate u iw Mw Crvo jru^iodi ly&l'Z «m,A 1905 '■9 ts 1-7 1.6 1.5 14 13 1.2 "*-.__ V 10° 20° 3 0° 40° SO" 60° 70' 80° 9 0° observations into two groups in such a manner that the heliographic latitudes increase with the position angle in one group, and decrease in the other. The error would thus show itself by the fact that the same values of the velocity would not correspond to exactly the same latitudes in the two groups ; or, to put it in other words, that corre- sponding latitudes should show slightly different values of the velocities. I have made such investigation for the whole time from 1901 to 1903, with the following result : — Table II. Heliographic Group I. Group II. Latitude. km. km. 82-5° 0-21 0-22 75-0 0-34 0-41 65-0 0-65 0-64 55-0 0-92 0-91 45-0 1-18 1-17 35-0 1-43 1-42 25-0 L64 1-63 15-0 1-77 1-79 5-0 1-89 1-89 SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 99 With the exception of latitude 75°, the differences between the two groups are far within the limits of their probable errors. This shows, as was anticipated, that practically no error of the position angle was present. We have now to deal with the question, How are these remarkable differences between the two groups to be interpreted ? Is their cause indeed to be sought for in the sun itself? We are, of course, not yet in a position to attempt an answer to this question, but there is a probability of the solar origin of this phenomenon, for which I should like to give some reasons. First, we have to take into consideration that the observations have been made throughout by the same observer, with the same entirely unaltered instrument, and at the same place, — that means, under the same atmospheric conditions. While it must be granted that deficiencies in the instrument and in the method of observation, or the personal perception of the line-displacements, which may be different for different observers, but most of all atmospheric conditions, which shall be discussed later — that all these circumstances may vitiate the results, there seems, however, to be no reason for the assumption that these influences should have altered so decidedly from one year to the next. Secondly, that a real change of the rotation must have taken place from the one group to the other seems to be indicated by a significant peculiarity of the angular velocities which have been derived from the linear velocities v contained in Table I. By multiplying the values of v by sec. /3, /3 being the heliographic latitude, we obtain numbers which obviously must be proportional to the angular velocities. This computa- tion having been made, the values were plotted down in fig. 4, again for both groups separately. At first glance we recognise the retardation of the angular velocity from the equator towards the poles, but it will be remarked that the amount of this retarda- tion during the second period is considerably smaller than during the first. The most significant fact, however, appears to be this : if in both groups the values of the curve for corresponding latitudes are subtracted from the equatorial velocity, these differences can be made to agree perfectly if the values for 1901-2 are multiplied by the factor 0'4. This result seemed to me so remarkable that I decided to test its correctness by a special investigation, including also the results of Professor Duner's observations. First of all I endeavoured to find an empirical formula which should represent the angular velocities in every group in a satisfactory manner. After various attempts at representing these velocities in the usual way by the sine and cosine functions of the heliographic latitude, the method had to be abandoned, as my observations could be represented only by extremely complicated expressions of such a form. Accidentally, however, I arrived at a formula very different from those hitherto used, which satisfies the observations in all three groups with a high degree of accuracy, and which has the further advantage of being extremely simple. This formula can be expressed in the following way — £ = a — bc $ , where £ is the angular velocity in latitude /3, and a, b and c are constants. When 100 DR J. HALM ON computing the constants of this entirely empirical formula, I found that c had the same value for all the three groups, but that a and b showed considerable differences. The value of c was found to be 1'01447 if /3 is expressed in degrees, and 2'2784 if expressed in units of the sun's radius. The constants a and b in each of the three groups are given in the following table, which also contains the comparison between the observed values of v and those computed by means of the preceding formula. Table III. Group a b 1887-9 (Duner) 2-349 0-354 1901-2 (Halm) 1903 do. 2-292 2-066 0-370 0-148 Group 1887-9. 0-4° v (comp.) 1-99 km. v (observed) 1-98 km. obs. — comp. -o-oi 15-0 1-85 1-85 00 30-0 1-56 1-58 + 0-02 45-0 1-19 1-19 o-oo 60-0 0-76 0-74 -0-02 74-8 0-34 Group 0-34 1901-2. o-oo 1-6° v (comp.) 1-908 v (observed) 1-908 obs. - comp o-ooo 5-2 1-881 1-894 + 0-013 8-3 1-851 1-871 + 0-020 12-8 1-797 1-802 + 0-005 18-0 1-720 1-720 0-000 23-8 1-617 1-594 -0-023 30-6 1-474 1-488 + 0-014 38-7 1-281 1-265 -0-016 47-3 1-055 1-061 + 0-006 55-8 0-822 0-840 + 0-018 65-1 0-566 0-560 -0-006 75-5 0-299 Groui 0-307 1903. + 0-008 P 3-7° v (comp.) 1-906 v (observed) 1-898 obs. - comp. -0-008 9-1 1-874 1-883 + 0-009 15 "5 1-813 1-831 + 0-018 21-1 1-741 1-753 + 0012 27-7 1-633 1-631 -0-002 34-0 1-513 1-512 -o-ooi 40-3 1-374 1-365 -0-009 47-1 1-209 1-207 -0-008 54-7 1-008 1-014 + 0-006 62-3 0-792 0-797 + 0-005 69-2 0-592 0-564 -0-028 76-0 0-393 0-408 + 0-015 830 0-192 0-187 -0-005 SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 101 The probable value of a difference " obs. — comp." is in all the groups approximately the same, viz., ±0 - 010 km., whereas the probable errors of an observed single value of v are respectively ±0*013, 0'013, 0"018. The formula therefore represents the obser- vations in all the groups with sufficient accuracy. The constancy of c is a somewhat significant feature. If it should be confirmed by future observations, it would almost certainly point to the conclusion that the formula is not purely empirical, but has some physical meaning. A decision on this question would be highly important, for if the rotation of the surface layers be actually governed by a law which has its analytical expression in the formula here adopted, we may conclude that surface phenomena are the cause of these peculiarities. I shall not dwell, however, on this at present, but shall now draw attention to the result of these computations with regard to the constants a and b of our equation. Here we find a satisfactory agreement between the values for 1887-9 and 1901-2. These are also the two groups which have exactly the same position in the sun-spot cycle, both embracing a time of minimum activity. On the other hand, as already pointed out, the year 1903 was characterised by vigorous displays of spots and solar eruptions. Simultaneously with this activity we notice an enormous change in the values of the constants of our equation. The " retardation " of the higher latitudes appears now to be reduced to less than half its former amount. This is a novel and, I am sure, unexpected result. So far, it is true, it can only be said to represent a coincidence in time, but I trust the discussion has made it sufficiently clear that the discrepancies observed, since they can in no way be ascribed to observational errors, must, in all probability, be interpreted as a new and very peculiar feature of the still mysterious mechanism of the sun. Care has been taken throughout not only to investigate all possible errors which may arise from deficiencies of the instrument, and to eliminate such errors from the reduced values of the observations, but the measurements were also arranged in such a way that the possible remainders of such errors which could not be elicited in the reductions should have a minimum effect on the results. The possible bearing of an investigation of this kind on important questions of solar physics makes it very desirable that the observations should not only be continued at this Observatory, but that other observatories should also take part in them. Next to the question of possible alterations in the amount of solar heat, an answer to which may now be expected from the ingenious researches of Professor Langley and his staff, the problem of solar rotation should command the greatest attention from the part of solar physicists, for no other seems to me so well adapted to give us information on the mechanism of the solar forces. That the periodic play of these forces should in some manner be conducive to changes of the distribution of the moments of rotation is a logical conclusion on which I trust astronomers are unanimous. It is well to remark in this connection that the idea has already received theoretical consideration in an in- teresting paper by Mr Emden. The construction of the apparatus required for these observations is well within the 102 DR J. HALM ON reach of our large astrophysical observatories, and there are several beautiful helio- meters which might be resuscitated for this purpose from their present unprofitable state of repose. Nor is the present the only investigation which can be taken up with an instrument of the kind here described. The construction of the apparatus, combined with its higli dispersive power, makes it possible to separate by a mere glance the telluric lines from the solar ones, and at the same time to determine by a single observation the wave-length of a line with an accuracy of about ±0'05 tenth-metres. The instrument appears therefore to be particularly adapted for an investigation of the telluric spectrum in accordance with the method first suggested by Cornu. As regards the accuracy of the determination of the rotational velocities, I am confident that, under more favourable atmospheric conditions, the annual output of observations can be considerably increased, and thereby the probable error of the annual means correspondingly lessened. It does not seem to me impossible that in this way changes of only one to two hundredths of a kilometre could be traced with certainty. I must not conclude this paper, however, without drawing attention to an incon- venience encountered in this spectroscopic determination of the rotational velocity. AVith a hazy atmosphere, a not inconsiderable quantity of scattered daylight, reflected from the particles of the aqueous vapour contained in the air, is thrown upon the slit of the spectroscope. As a certain percentage of this light will have emanated from the interior of the sun's disc, the solar lines of this " day " spectrum appear less dis- placed than the lines of the two solar limbs. By a superposition of the two spectra, the intensity of the lines will therefore be lessened on the one side and increased on the other, the effect always being to bring the lines nearer to their normal position. The velocities obtained under such conditions must therefore be too small. I have often had occasion to observe this phenomenon when, after a measurement made under favourable atmospheric conditions, the sky was suddenly overspread with a veil of haze, a meteoro- logical feature not at all infrequent in the climate of Edinburgh. When the observation was then repeated, the measured displacement was invariably found less than that obtained before. Observations under misty conditions of the sky were therefore care- fully avoided. A reliable scale for the transparency of the atmosphere was supplied by the optical appearance of the dark band separating the spectra of the two limbs in the viewing telescope. Whenever this space became so bright as to show traces of the absorption lines of the solar spectrum, the observation was broken off. All the measure- ments used in this discussion were made at moments when the band mentioned was uniformly dark, so that the spectrum of the diffused daylight cannot have seriously influenced the measured displacements. SPECTROSCOPIC OBSERVATIONS OF THE ROTATION OF THE SUN. 103 Table IV. Single Values of Linear Rotational Velocities of the Sun observed during the Years 1901-1903. Lat. Vel. 1901. Aug. 13. 24-2 1-41 381 50-4 64-1 77-0 79-2 64-6 48-5 32-9 17-0 Aug. 23-0 Aug. 8-5 5-6 19-8 33-3 47-6 60-7 74-0 Aug. 11-5 2*5 16-7 30-8 44-5 58-3 71-8 779 65-0 51-0 37-1 23-6 9-4 4-5 Aug. 1-0 2-2 11-4 25-2 39-0 52-8 665 4-5 5-5 21-1 Aug. 2-3 1-23 0-84 0-55 0-22 0-12 0-71 0-97 1-17 1-49 15. 1-52 16. 2 1 1 1 1 19 1 1 1 1 1 1 1 1 1 1 20 2 1 1 1 1 1 1 1 03 77 76 45 17 67 34 70 85 60 23 20 57 16 33 51 08 28 43 48 75 03 80 59 39 08 84 32 93 80 52 21. 1-79 Lat. 3-4 5-5 14-9 23-7 2-3 31-4 40-4 49-3 58-4 67-2 75-9 5-5 10-5 Vel. •69 •75 •65 •58 •75 •31 •13 •83 •66 •35 •32 •77 •83 Aug. 22. 0-8 2-1 8-2 1- 17-0 1- 25-5 r 34-3 r 43-2 i- 52-2 i- 6M o- 69-7 o- 5-4 2- 6-6 2- •07 •87 •84 •83 •46 •19 •03 •75 •42 06 •11 Aug 37 5-0 3-4 11-1 20-5 29-1 34-9 44-2 Aug. 78-6 68-8 58-1 47-4 36-7 25-8 14-6 3-8 7-2 7-9 26-7 17-8 24. •06 •11 •96 •82 •67 •54 •20 •08 25. 0-11 0-52 0-89 1-19 1-41 1-64 1-87 1-98 2-05 2-04 1-84 1-86 Aug. 27. 6-6 2-00 8-1 1-85 Lat. 0-5 16-9 Vel. 2 05 2-09 Aug. 28. 10-0 1-89 18-6 1-92 27-4 1-83 36-3 1-59 43-5 1-33 52-4 1-07 51-0 0-72 1-1 1-90 Aug. 30. 193 1-70 2-5 1-88 8-3 1-89 Aug. 31. 1-9 2-17 1-1 1-95 Sept 1. 6-5 1-90 19-3 1-97 10-3 2-03 1-8 2-13 70 2 09 15-9 1-98 Sept. 50-6 59-2 68-0 2-2 8-1 1-0 9-5 18-3 27-0 2. 1-03 076 0-32 1-94 1-88 2-17 1-92 1-83 1-59 Sept. 4. 28-8 i-53 17-4 1-71 5-7 2-03 5-9 2-21 Sept. 5. 0-8 2 02 Sept. 7. 7-7 1-85 41-1 1-34 48-9 1-19 Lat. Vel. Sept. 10. 62-9 0-77 52-2 41-4 30-8 19-7 8-7 Sept. 15-9 24-4 33-3 42-4 51 3 60-0 68-6 Oct. 69 18 10-3 18-9 Nov. 39-6 48-2 56-3 64-6 73-2 1-2 9-0 16-8 1-04 1-20 1-59 1-87 1-92 11. 1-99 1-77 1-67 1-37 1-00 0-87 0-56 26. 1-98 2-11 1-84 1-49 5. 1-30 0-99 0-86 0-63 039 2-00 1-73 1-67 1902. Feb. 12. 13-0 1-72 5-3 2-7 10-5 189 1-89 2-01 1-73 1-68 Feb. 13. 7-9 2-14 Feb. 15. 33 1-85 14-6 26-1 37-2 48-9 60-7 71'4 1-79 1-46 1-07 0-98 0-77 0-42 Lat. Vel. Feb. 67-8 55-0 Feb. 35-7 27-1 18-9 10-3 1-8 6-8 14-8 23 31-5 39-8 Mar. 46-0 54-5 63-0 7L4 Mar. 76-2 65-9 54-7 43-3 32-3 13-9 2'9 Mar. 21-0 9-2 1-8 26-7 35-5 43-9 51-5 17. 0-31 0-63 18. 1-39 135 1-46 1 71 1-67 1-74 L75 1-67 152 1-14 7. 0-96 0-78 0-51 035 15. 0-23 0-55 0-88 1-25 1-43 1-90 2-05 16. 1-64 1-90 1-89 L47 1-22 1-04 0-84 Mar. 20. 9-5 1-77 6-8 1-84 18-1 1'76 Mar. 21. 17-6 1-66 Mar. 6-5 0-2 7-6 16-0 24-3 22. 1-85 1-77 1-90 1-83 1-66 Lat. Vel. 32-1 1-54 40-7 1-32 49-2 0-99 Mar. 24. 62-0 0-64 Mar. 25. 70-6 0-44 59-8 0-75 48-8 1-10 37-8 1-28 26-4 1-61 15-5 1-90 4-2 1-96 7-3 1-85 Mar. 29. 21-1 1-58 Apr 6-7 . 1. 1-89 Apr 0-4 2 1-77 8-4 1-89 16-5 1-75 25-6 1-50 31-4 1-48 40-1 1-14 Apr 50-1 .7. 0-98 Apr. 52-6 10. 1-04 61-5 0-86 70-6 0-67 78-5 0-16 Apr. 79-5 16. 0-24 71-1 0-46 75-9 0-13 655 0-61 54-9 0-85 44-3 1-14 33-6 1-46 2L8 1-55 10-8 1-63 0'4 1-83 Apr. 5-1 18. 1-85 16-0 1-80 Lat. Vel. Apr. 25. 18-0 1-61 27-3 1-73 Apr. 27-9 17-1 6-4 5-0 16-1 27-5 Apr. 18-1 28. 1-72 1-66 1-84 1-74 1-47 1-51 30. 1-78 May 29. 0-6 1-92 14-9 25-5 36-5 47-0 57-6 68-7 1-87 1-41 1-28 107 0-82 0-64 June 27. 52-7 0-69 63-0 73-8 44-2 34-7 25-2 15-5 0-43 0-22 102 1-30 1-53 1-76 June 30. 10-4 1-79 10 8-8 18-0 27-3 36-8 46-2 55-8 July 3 8 5 1 6 1-88 1-75 159 1-48 113 0-93 0-63 5. 0-70 0-98 1-23 1-41 1-66 1-73 1-85 2-00 1-76 1-39 104 DR J. HALM ON SPECTROSCOPIC OBSERVATIONS OF THE SUN. Table IV. — continued. Lat. Vel. 1902. July 5 44-2 54 4 649 65 3 75-5 contrf. 102 0-83 0-58 0-56 0-29 Aug. 21. 49-8 1-00 Aug. 59-8 50-2 41-5 33-0 23-8 14-7 Aug 5-9" 2-5 10-2 17-7 23. 067 0-92 1-16 1-23 1-49 1-73 25. 1-71 1-54 1-75 1-66 Aug. 26. 35-6 24-9 11-2 0-7 22-4 3-6 37-1 •41 •65 •65 •73 ■69 •78 •44 Aug. 27. 6-4 1-72 Aug. 28. 8-2 1-92 17-2 25-7 34-8 43-6 52-8 61-5 70-5 78-7 78-4 1-85 1-66 1-44 1-31 0-96 0-69 040 0-15 0-37 Aug. 29. 74-4 0-41 Lat. Vel. Sept 79-1 . 8. 0-39 74-2 0-13 64-0 0-63 535 0-89 42-9 114 30-4 1-40 19-7 1-65 8-9 180 Sept 18-6 9. 1-72 8-0 1-86 Lat. Vel. Sept. 13. 2-9 1-81 11-0 1-86 Sept. 17. 12-2 1-90 1-4 2-01 9-4 1-85 Sept. 9-9 1-0 6-9 15-1 21-1 29-8 23. 1-80 1-95 1-89 1-77 1-71 1-52 Oct. 4. 36-7 1-50 45-5 1-17 54-5 1-08 62-4 0-71 70-8 0-33 Oct. 19. 21 1-70 4-5 1-89 13-3 1-76 21-9 1-64 30-7 1-50 39-4 1-17 48-3 1-02 Oct. 62-1 70-4 78-3 79-0 30. 0-57 0-29 0-13 0-48 Nov. 80-0 72-0 Nov. 7 1. 0-41 0-53 4. 0-84 1-05 1-32 1-54 1-64 1-61 Nov. 10. 777 0-29 1903. Feb. 25. 24-0 1-65 38-8 1-30 Feb. 27. 35-4 1-52 17-2 1-87 9-4 1-94 Mar. 14-5 2 3 8-9 20-5 31-8 52-8 63-1 15. 1-83 2-02 1-99 2-00 1-68 1-19 0-87 Apr. 17. 7-2 1-85 18-3 1-78 29-3 1-65 Apr. 16-3 28-3 Apr. 9-9 20-8 31-6 41-9 530 64-0 74-3 83-8 81-7 71-5 19. 1-79 1-60 24. 2-13 2-01 1-70 1-34 1-03 0-74 0-42 0-16 024 0-53 Lat. 59-7 48-6 May 3-9 7-1 17-8 28-5 38-7 49-0 59-7 70-9 81-3 74-8 64-3 53-7 43-1 32-5 May 46-6 36-3 25-9 15-0 3-9 6-8 173 27-9 38-4 48-9 59-6 69-9 May 64-8 75-3 86-3 83-1 72-9 61-5 51-0 40-5 28'8 18-4 7-7 2-8 13-1 Vel. 0-81 1-03 23. 1-84 1-86 1-80 1-65 1-29 1-17 0-77 0-53 0-26 0-48 0-75 1-04 1-35 1-46 26. •25 •41 •69 •77 ■91 •83 •93 •74 •54 •20 •99 •63 27. 0-89 0-56 0-19 0-14 0-33 0-79 1-02 1-35 1-61 1-68 1-84 1-91 1-93 May 28. 4-4 15-1 25-9 36-1 47-3 57-6 68-1 •93 •86 •66 •46 •26 •95 •53 Lat. Vel. Ma 1 { 29. 72-8 0-49 83 3 0-24 85-9 -0-14 75-8 033 65-4 0-68 54-8 0-82 44-2 1-13 34-0 1-56 235 1-73 13-2 1-78 2-9 1-84 7-7 1-73 June 3. 5-5 1 82 4-1 1-84 13-5 1-67 June 12. 20-6 1-95 30-1 1-63 39-6 1-31 June 20. 33-3 1-44 42-9 52-5 62-1 71-5 81-2 79-2 69-9 1-26 1-07 0"77 0-33 0-09 0-55 0-52 Aug. 14. 41-4 1- 31-9 1- 23-0 1- 135 1- 4-4 1- 5-1 1- 14-1 1- 23-4 1- 32-3 1- Aug. 40-0 48-1 57-4 66-1 74-7 81-2 75-5 Aug. 69-2 60-5 46 41 69 88 98 90 96 81 67 17. 1-68 1-34 1-10 0-75 0-58 0-31 0-14 18. 0-39 0-65 Lat. 51 2 42-3 32-9 23-5 24-5 Vel. 0-92 1-13 1-47 1-56 1-68 Aug. 27. 1-9 1-81 Sept. 70-9 78-8 Sept. 16-7 30-0 38-4 44-4 53-7 14. 0-56 0-09 15. 1-59 1-36 1-28 1-28 1-00 Sept. 18. 73-5 0-73 64-0 54-9 46'0 37-6 28-4 19-3 10-0 0-2 Sept. 10-9 19-4 28-6 37-2 46 3 52-5 60-1 68-8 77-3 Sept. 66-9 Oct. 67-5 75-0 81-8 80-5 Oct. 66-0 78-7 836 79-8 0-92 1-13 1-17 1-57 1-73 1-72 1-93 1-96 19. 1-92 1-66 1-58 1-33 1-20 1-20 0-87 0-42 0-23 27. 0-64 11. 0-76 0-33 0-28 0-20 13. 0-78 0-32 0-16 0-46 Lat. 70-0 59-1 47-2 35-9 251 Oct. 18-1 6-4 5-5 17-4 Vel. 0-55 0-71 1-20 1-45 1-67 21. 1-90 1-94 1-83 1-84 Oct. 23. 11-2 1-94 3-2 1-85 Oct. 11-9 20-7 29-3 37-6 42-5 50-6 58-5 Oct. 67-2 56-3 44-3 33-4 21-3 9-8 Nov. 6-8 15-6 23-9 32-4 40-4 48-8 57-2 27. 1-79 1-72 1-61 1-42 116 097 0-77 31. 0-68 0-94 1-28 1-49 1-51 1-82 3. 1-82 1-89 1-66 1-57 1-38 1-11 1-04 Nov. 4. 3-1 1-99 12-0 1-85 20-5 1-82 27-1 1-60 Nov. 41-0 49-9 58-1 65-4 73-8 81-6 86-0 6. 1-57 1-28 1-01 0-69 0-57 0-33 0-16 ( 105 ) VI. — Theorems relating to a Generalisation of the Bessel-Function. By the Rev. F. H. Jackson, H.M.S. " Irresistible." Communicated by Dr W. Peddie. (MS. received February 17, 1904. Read March 21, 1904. Issued separately May 27, 1904.) 1. In this paper, theorems which are extensions of the following, are discussed : — J (x + y) = J (x) J (y) - 2J 1 (x)J 1 (y) + 2J 2 (x)J 2 (y) - ad inf. . . . (a) 1 ={J {x)}*+2{J i (x)}* + 2{J s (x)}* + adinf (/3) t i \ i i\»It a„\ 2m(m + n) T . . , 2 2 m(m - l)(m + n)(m + n - 1) T , N | , > \ X 2] x z I S_ n (x) = (-lfJ n {x) t / \ / ^^n{^) n+t d n (sin a;) ,,. We define J [n] (A, x) as r=oo ^ £>[r~]\[n + r}\{2) r {2) n+r In this expression [n + r]\ is T p ([n + r + 1]) or II p ([> + r|) r -, • B" - 1 ^" + " L ^ J iy[»+7TI]) The function IIp([n]) is defined in the previous paper (Trans. Roy. Soc. Edin., vol. xli. part i.). If A be changed to iX in J [n] , the function will then be more strictly analogous to J n . In Weierstrassian form — ~ = \xy x "u i f l + »-*m V^ ] !■ r p ([a-]) L J „=i I \ 1 [n]J J P=l +r l + -L+ -log--? [3] T [S] >-l {2n}l-(2),,r,([ n .+ l]) being in the case of n positive and integral {2»}l-[2][4][6] . . . [2n] analogous to 2 . 4 . 6 . . . . 2n = 2".n\ This notation enables us to write shortly l lV»'\"+2r d[n] 2 J {2« + 2r}!{2r}! TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 6). 106 THE REV. F. H. JACKSON ON 2. If we invert the base element p, we see that [r]! is transformed intop"'' (r " 1)/2 [r]! and that (2) r becomes transformed into p' r(r ~ 1),2 (2) r . These transformations hold whether r be integral or not. Inverting the base p in the series J [n] ft), we obtain „«*v (-W (n+r ' AY +ar a) P H[r]\[n + r]l(2) r (2) n+ XpJ ' ' ' ' {> which we denote y 2 i M (|) (2) Lommel has shown that 'm + n + 2r' <wua) = 2/-i)"_ _i_ _2 (J) ... (3) '•=" T(m + r+l)T(n + r+\) The function J [n) was formed while seeking to extend the above theorem. The extension was found to be JwWImW- JwWiwW- Z. ol>([m + re + r+ l])r p2 ([m + r+ l^r^n + r + l])r p2 ([r+ l])l [2]^ V ' The relation between J and J was surmised from the following simple but similar theorem. 4W - 1 + rTT, + m + [1]! [2]l ■lW - i + £p + ^ + w« - > - S + -pf + which suggested that J| [n] might be derived from J [n] by inverting the base p. As I have given the proof of (4) as an example of the use of generalised Gamma-functions in a paper communicated to the Eoyal Society, London, it will be sufficient to say here, that the theorem may be proved by showing, that the coefficients of the powers of X on both sides of the equation are identical, being cases of the extension of Vandermonde's theorem (Proc. Lond. Math. Soc, series 2, vol. i. p. 63). In the notation of Art. ( 1 ) we may write the theorem JwWImW = ^{2m + 2?i + 2r}\{2m + 2r}l{2n + 2r}\{2r}\ Xm+ " +r ' ' ( 5 ) Consider the series •i l n l (A)|„ 1 A) + fflj m (x)| [1] (A)+ +jp s - 8 - 1 [Sj w (X)I w W+ (6) THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. 107 By means of (5) we write this — h • MM. >2 + ( l)r W X «r + \ I [2] [2] [2] [2] {2r}!{2r}!{2r}!{2r}! + J [2] I [4]"[2] [2] [2] {2r + 2}l{2r}l{2r}l{2r-2} i w 9 r.MMj t4r^ A2r + [2r] I i4r}l{2r}I{2r>I{0}J We see by inspection that the coefficient of A 2 vanishes : the coefficient of A 2 '' is the expression n y W h_W J2d_ + «J23 [2r][2r-2] V ; {2r}!{2?-}!{2r}!{2r}! I [2] [2r + 2] 1 [4] [2r + 2][2r + 4] ' ' ' ,/ 1)y M,M [2r][2r-2] [4] [2] \ ( + { )P [2r][2r + 2][2r + 4] [4r] j ' K> The series within the large brackets is easily summed term by term. The sum of the first two terms is —p%— i, which is a factor of the third term. The sum of L [2r + 2] — ttz d. and the third term is ^[2r + 2] [2r-4] [2r-2] jP [2?- + 2][2r + 4] Continuing in this way, we obtain that the sum of the first r terms is / _ nr - 1? ^- 1 , [2r-2][2r-4]l2r-6] [4] [2] V ' l [2r + 2][2r + 4][2r + 6] [4r-2] which is equal to the last term, but is of opposite sign. The coefficient of X 2 '' is zero, and only the constant unity is left. Therefore 1 = J[0](X)l ra (A) + t|jj [1] (A)l w (A)+ +P"- 1 |JJ [ #)| 1 #)+ (9) which, if p = 1, reduces to 1 = {J p + 2{J i p + 2{J 2 P+ (10) 4. Consider now the series JraMfmW-[^J w (A.)I ra (A.)+ +P r ' r -^MM*) - (H) Referring to expression (8), we see that the coefficient of A 2 " is (-If, , , W . [4] _[2jj_ [8] [2r][2r-2] , V ' {2r}!{2r}!{2r}!{2r}!} X + [2] [2r + 2] + ^[4] [2r + 2][2r + 4] J V ' 108 THE REV. F. H. JACKSON ON The series within the large brackets, although simple in form, offers considerable difficulty in summation. The sum is 2(p» + l)(p*+l) Cp*-*+l).(p» + l)(p*+l) . . . (^+l)l^iMl. . (13) The sum in general for all values of r is [2f [4r] r, 2 ([2r + 2]) ' ' ' { > The reason that the simple series (p= l) 4 _2r_ 8. 2r(2r-2) 2 2/- + 2 4 (2r + 2)(2r + 4) is easily summed as 2 "2* while the general series offers difficulty, is that the functions T p are present, both to the basep 4 and the base_p 2 in the series (12) and in expression (14). Heine has shown in his " Kugelfunctionen" that ^,b,c,p,x\-l + x+ x + „_.<! - bxp n )(l - p>») = n 6 ,bx,p,- e b (15) n=0 (l-»p»)(]-cp») ■Consider now the series [T][r+I] + ^[2][r + lJ[r + 2] + * [.] [r + 1] [r + 2] . . . [r + s] + ' " ' ' (lb) Since [1] i?-l W =^zi =^ + l [2] J5 2 -l ^ we write series (L6) as the sum of two Heine's series b i + ^+i _ 1 & 2 | + jfH _ j +^ r+I _ 1)(pr+2 _ l} + ■■ ■ ] V +1 - 1 ' f +2 - 1 (y +2 - 1) . . . ('P r+s+1 -i)i • ( 7) These series we transform by means of (15). First, for the series S x we put a=p b=p~ r c=p r+1 x= -jf THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FTJNCTION. 109 and obtain, after obvious reductions, _ 2(1 + P )(l +p*) (1 +p*->) j P * - 1 p (p*-l)(p*-*-l) P * (1 -y +1 )(l ~P r+i ) ■ • ■ ■ (1 ~P 2r ) t P-I 2 (i?' 2 -l)(^ 2 -l) 2 (p»-l)(p*-*-l)(p*r-*-l) pj ) (lg) (j> 2 -l)(l> 8 -l)(p 4 -l) 2 ■•■■[ In the same way, if we put we obtain, after reductions a=p b = p l ~ r c=p r+2 X= — p r+1 s,-i + ^:^ + v r - 1 -i , — 7-S P ' p r+i - r +ff 2 )(i +p 3 ) (i +p r ) / 1 . p 2 "' 2 - 1 J: , 3 , (p 2r ~ 2 - i)(^ 2r - 4 „ 8 _ I (i9) (i-^ +2 )(i-y +3 ) (i-p 2r )t (p-i)(p 2 +ir (p-i)(p 2 -i)(p 2 +i)(p 3 +ir ""< T 1 So that S, +p p ~ S 9 may be written p' + -1 " { i+p)(i+p*) (l+y 1 ) ( „ _.P 2r -i , (p 2 --i)(/ r - 2 -iU _ (l -^ +1 )(i -y+ 2 ) .... (i -i? 21 ) I p - 1 7 (p 2 - 1)(/> 2 - 1) p*-l (p 2 --D(p 2r - 2 -l )„4_ I . (20) p+1^ (P 2 -I)(i> 2 +1) l J Adding the terms with like numerators together, we obtain (1+P)q+P 2 ) (1+P'- 1 ) I 2 _ 9b2 /^l . o, yB (P 2 " - l )(f r ' 2 7 !) _ 1 (21) (l -p r+1 )(l -^ r + 2 ) (l -_p*) I " ^ r - -I 2 (p 2 - l)(p 4 - 1) ' ) The series within the large brackets is the simplest type of series, and its sum is well known to be 2(1 -jj»)(1 -p+) (1 -jF) .... (22) We have therefore i + EIJzL. + J£ Mfr- 1 ] + [!]['■+!] f 2 ] [' + !]['■ + 2] " ' " 2(1+^(1+^) . , , (1+J/-1) • (1 -^1-^) . , . {l-p*) (l-;/+ 1 Kl-i/ + --') (l~P 2r ) Changing the base p to p 2 we obtain the series whose sum was sought 1 + [*] JM. 2 [8] [2r][2r-2] [2][2r + 2] ^[4][2r+2][2r + 4] = 2(l+jJ 2 )(l+j? 4 ) ■ . (1+jr"- 2 ) • (l-p 4 )(l-p 8 ) ■ • . (l~P 4r ) (1 -y +2 )(l -p- r+i ) (1 -p ir ) (23) (24) The coefficient of \ 2r is obtained by multiplying this sum by W {2r}! {2r}! {2r}\ {2/-}! 110 THE KEV. F. H. JACKSON ON which gives us nr 2(l+^)(l+y)(l+y«) . . . (1+F- 2 ) • (l+j>*)(l+^) (1+^Q (25 , [ ' {2r}!{2r}! ' ' V ; as the coefficient of A 2r . If r be not integral, the infinite products in Heine's trans- formation do not reduce to finite products but to expressions in terms of the F p functions, ultimately giving the sum of the series in the form (14). Having obtained the coefficient of A 2r , we have established, subject to convergence, the theorem JraWIraW - [2]j w (^)I [ u(^)+^ 2 [|jj [2 ](^!m(^)+ _ 2(1 +p*)\? ( iy . 2(l+^) . . . (l+y- 2 )-(l+^) . . . (l+f)X» , " " {2}!{2}! + ( } {2r}!{2r}! " (26) which is the extension of J (2X) = {J (X)P-2{J 1 (X)P + 2{J 2 (X)P- (27) This is a particular case of the addition theorem for J . J (A + A 1 ) = J (A)J (A 1 )-2J 1 (A)J 1 (A 1 )+. . . . 5. Defining * Sin p (A) and Cos p (A) as Sin„ W = X-| !+ | r .,.. we obtain Sin, (A) Cost (A,) + Cos, (A) Sum (A,) = (A + A,) - (^ + W + V 2 ) (* + V 4 ) + .... p p [dy. Sin, (A) Cos x (A) + Cos,,(A)Si ni (A) = 2A _ ^(1 +^)(1 +i^ 3 + p p [3J! This suggests that the extension of the addition theorem of J (^ + \) will be on similar lines. Consider now the series JraWI ra (A 1 )-||jj w (A)| [1] (A l ) + (-l)'y"'- 1 'Hj [ i](A)l w (A 1 )+ (28) I'ioc. Ed,in. Math. Soc, vol. xxii., 1904. THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. Ill and the product J[»](^)Im(\) - J x n ^ . , ' *l +2r _ U V - + K +2r P 2r(n+ r) _ I m s I {2ra}! {2ra + 2}!{2}! '" {2n + 2r}\ {2r}\ ' ' ' I I {2n}\ {2n + 2r}\ {2r}\ ) K ' From series (28) we are to form a new series, of which the successive terms will be homogeneous in AX X and of degrees 0, 2, 4, 6, 2r, . . . , . respectively. The first term of (28) gives rise to the constant, unity. The terms of the second degree arising from J [0] J [0] are A 2 A % p 2 "{2]TT2}! ^ "{2M2}! The term of the second degree arising from J[i]Jf[i] is [2] {2}! {2}! There are no other terms of the second degree ; the sum of these terms is _ (A + y 2 )(A + A 1 ) {2}! {2}! Terms of the fourth degree arise only from the first, second, and third terms of (28), heing respectively a 4 A^y \y {4}!{4}! + {2}! {2}! {2}! {2}! {4}! {4}! [£U *%__ + A W - I . . (30) L2]}{4}!{2}!{2}! {4}!{2}!{2}! f * > y [4]{4}!{4}! Remembering that {4}! = [41[2] and {2}! =[2] we write the sum of (30) A4 . M l^)\3\ . W_W>2\ 2„2 , [S]\2A 2 W 2 4. [ilM A A 3 4. „tS\ i \ [4] [2] [4] [2] 1 '[2] [2] 1T [2][2] ^'[4] ^'[2] [2] Replacing W by (jps + l) and ® by p* + l the expression within the large brackets reduces to (A + A 1 )(A +p%)(\ +p%)(X+ P %) The term of the sixth degree in X, \ I have verified as ~ L6J L4] [2] 1 [6] [4] [2 ] { (A + Xl)(A + X ' p2){X + V * } * (X + Al?,2)(A + Xlpi){X + Xip6 } ' (31) 112 THE REV. F. H. JACKSON ON The term of degree 2r is the following expression — I |2r)!{2r}! |2r - 2}!{2r - 2}!{2}!{2}! I2r}!{2r}! 1 + Wi A 2 '-^, A'-'-y^ [2]( {2r}!{2r-2}!{2}! {2r-2}!{2r-4}!{4}!{2}! A 1 2r -'Ap 2 '-"- 1) +_p : 2 [s]( A*--y ., .+ A'- v -y/^ [4] I {2r}!{2r-4}!{4}! {2r- 2}!{2r- 6}!{6}!{2}! . . . + {2}!{2r-2}!{2r} {4}!{2r-4}!{2r}! J M! 1 • (32 > [4r]f __Ay_) ^ f2rl U2r}!{2r}!j [2r] I {2r}!{2?-}! We have shown in Art. (4) that in case A = X 1 this expression is (A + A)(A + Aj) 2 )(A + Aj; 4 ) . . . (A + Ajr'- 2 )-(A + Ap 2 )(A + A^) (A + Ajr r ) {2r}!{2r}! (33) It has been directly verified that for particular values of r (l, 2, 3) the forms, in case X be not equal to \, are _ (A + A 1 )(A + A 1 p 2 ) {2}!{2}! (A + A t )(A + A lP 2 )(A + A^)(A + y*) {4}!{4}! _ (A + A, )(A + A ^XA + \ lP *) ■ (A + A 1 j/)(A + A^)(A + A 1 p«) . {6}!{6}! respectively. This indirectly establishes the form of the coefficient of degree 2r in A and \. A direct proof of the algebraic identity would, however, be preferable. Writing now £f >F j^^WAi) - i + Z.V-^ {2«}!{2w}! (34) If p = 1 , we obtain the addition theorem of J JofA + Aj) = J (A)J (A 1 )-2J 1 (A)J 1 (A 1 )+ , . . (35) 6. The analogue of Lommel's theorem j.w = (-ir|j^)- 2 ^ ) w 1 w+ } I have shown by two distinct methods * that t (py~«- l){ 2)y~ a+1 - 1) . . . (py- P +'-i- l). (py-P-l)(py-P+ l - 1) (jjy-£+«-i- l) K ii (^y-«-/»-l)(pv-*-0+ 1 -l) . . . (#-■- M-«-i-i)- (pv-l)( j ijr+ 1 -l) . . . (pv+*-i-l) p Ii][y] ' [iJ^]M[y+i] (36) * Proc. Lorad AfaiA.. Soc, series 2, vol. i. pp. 71, 72, 1903, and Amer. Jour. Math., vol. xxvi., 1904. THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. 113 In terms of the function T p of this paper, this theorem is J_ iy[y-a-/3] )iy[ y ]) = [a]j£j ^ r y ([ 7 -a])r4 7 :/i]) + ^[i][y] " " " " ' ' (6i} Change the base p to p 2 and put a = — m f3 = -m-n y = r—m+1 we obtain IVflr + m + n + l])iy,([r - m + 1]) = 1 + [2m] [2m + 2» ] y2 _ 4ffl _ 2re ( I>(|> + l])r^([ r + »+!]) t 2 ] [2r - 2m + 2f [2m] [2m - 2s + 2]- [2m + 2m] [2m + 2n - 2s + 2] ^_ 2t(2OT+ „, ,„„, [2] [4] . . . [2s]-[2j--2m + 2] . . . [2r- 2m + 2s] F y ' Now consider . .. p» [2m ][2m + 2»] T ... f [2m][2m - 2][2w + 2»][2m + 2n - 2] T ,., , Wn]W "^ShS L ~ JL |- 2 j A J -J Pm +n-vW + pSm+in L j- 2 ][- 4 -| J «W.-«W + • ■ ■ ■ (39> The coefficient of X' l+2r is the infinite series Iz 1 )^'" [~i+ p2 [MI2^±M + 1 uo) {2n + 2m + 2r}l{2r-2m}\i p im+2n [2] [2r - 2m + 2] J ' " * y which by (38) reduces to (-1)'-'" I>([m + » + r + 1 ])r^([r - m + 1]) {2n + 2m + 2r}l{2r-2m}\ I>([r + l])I>([r + n + 1]) Now remembering {2s}! = [2]T i)2 ([,+ l]) = (2) s r,([.+ l]) the expression (40) reduces to r7([r + l])r,([ W + r+l])(2),.(2)„ + ,. (41). ■n (_iy-» P . . (42). which is ( - i)-»p2m ( m+n) x coet }icient of \ n+2r in the series J [n] . This establishes ( - iyy m,m+ "»j M (A) = j [2m+n] (A) - _£. C^[2^±2»] J[2m+B _ u(X) + — ad inl . (43) . an extension of j„ = (- ir < j»h. - 2 ^ ± zL ) J 2M+n - 1 + } • • (44). Lommel defined J n for negative integral values of n, so as to make this theorem always hold : for examp]e, suppose n a negative integer, and put it equal to - m, then we have by this theorem j [ _ n] = (-irj [n] (45> extending J_„ = (-1) W J„ also ![_„ = (-i)*l M as may be shown by inverting 'the base p in expression (43). TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 6). 18 114 THE REV. F. H. JACKSON ON 7. If we now define Sm p (\,x) as A* - _. + — _ (46) then Sm„(A,^) 1 _ A 2 a: [2] Operating on this with g [n] which is I d \2n+2r„.[2n+2r] K ' [2»+2r + l]l ' 1 d(x^) \ d{aF) ) ] ) (47) the first n terms of the series are destroyed, while the term involving xP n+2r] is reduced to T2r+2l o Taking , _ 1 x n+ r A 2«+2r 1 [2w + 2r][2« + 2r-2] . ■ . ^ ; [2]" [2n+2r + l]l (48) ^n+j+2r a ;[«+J+2»-] L« + r + J]![r]!(2) r (2), 1+9 . +i [n + r + fll = [» + , + *] [n + r-J] [f]r,([|] (2) n+r+} = d*-^» + l) ( 2 <-+'-»+ 1) .... (pf + 1) • (2), and therefore So we obtain J [n+i] {\,x) (2) i r,([i + |]) = [2]»r^([l+i]) [n + r + fi\(2) n+r+i = [2rc+2r+l] ,2(- 1 )"r, . . [3] x [2]iU P 4i]) [2]*T^[f])^- x '[2 W + 2r + l] . . . [5] [3] • [2] [4] . . . [2r] and by a change of the variable x A J [TO+H (A,a ) - ( 1) - [2]I>([f]) A 1 -j(F" J ' A w denoting the operator (49) (50) (51) (52) For further properties of Sin p , Cos^, and their connection with symbolical solutions of certain differential equations, reference may be made to a paper on " Basic Sines and Cosines" (Proc. Edin. Math. Soc, 1904). THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. 115 Continuation op Paper — " Theorems relating to a Generalisation of the Bessel-Function." (MS. received April 19, 1904.) 8. The theorem jmm»)-$m%m+ +(-i)y (s - 1, Bj]jc S] («)i M w- 1 _ (a + b)(a + bp 2 ) (a + b)(a + bp 2 )(a + bp 2 )(a + 5p 4 ) _ , , [2] 2 + [2] 2 [4p ~ - ' * {a) discussed in the first part of this paper may be obtained very naturally from the properties of a certain function analogous to the exponential function. Elsewhere," 3 " by means of the function E p I have obtained Wh(j)~WMh(j) + +(~mP*Jd*)h(j)- ~W~ Pit*? ' w We naturally expect to find some general form to which both (a) and (/3) will belong, as particular cases. The following is the general theorem which will be obtained from the function E p , just as the addition theorem for Bessel coefficients is obtained by means of the exponential function. Exp. ( ■xi t — - J ^(a.&J-J^a)!^^ (y) [2vJ |>J T" /„ h\ - l ( a + t>)(a + bf) (a + &)(« + ty 2 )(« + 6p*)(a + &p Sr+2 ) [2] 2 T " _ [2]W In case v = 0we have the quasi-addition theorem (/3). If, however, v= 1 we have the quasi - addition theorem (a). The corresponding theorems for the function J* (a , b) will be briefly noticed. y , h) _ (a + b)(a + bp*) .... (a + bp*-*) ( , _ {a + bp*)(a+bp>) °^ a ' 0) [2] [4] . . . [2ra] 1 [2» + 2][2] " (a + fy? 2 ")(a + bp 2n+i )(a + fb)(a + ;r" +2 6) _ I (S) + ~ [2n+2][2n + 4][2][4] ~ ' ' ' J * j The expression for J* (a , 6) will be given also in the case when n is not a positive integer. * Proc. Lond. Math. Soc, shortly to be published. 116 THE REV. F. H. JACKSON ON 9. In this article certain results will be obtained which will be required in subsequent work. We define the function E p (a) as E *)*- 1+ [T]! + r2]! + If we invert the base p E i«- 1+ [rp + *[J + ■ • • • + ""-'% + ■ ■ ■ ■ without difficulty we have E»EL(*) = l + (ffi + (« + »>fr + »> + (e) Changing p to p 2 M.»h(»)-i+^+^ X '^ + .... p 2 — 1 /o 2 - 1 p 2 — 1 _ [2](q + &) [2]*(a+&)(a + 6p*) [2] + [2] [4] "+••■• M[2]JM[2]J = 1 + -[2T + [2] [4] + (r?) In part (l) we have established J t _ n] («)= (-l)"J tB] (a) (6) Inverting the base p we obtain also from this l^/a) = ( - in M (a) ( K ) 10. A consideration of the product of the two absolutely convergent series w ( at\ , at u 2 t 2 . a"t n M[2]; = 1+ [2] + [2j[4] + ••■• + {m> + E A-[2]-;- 1 -[2T + [2][4]- •••• ( - 1) TM! + •••• shows us that E *(^) E *(-$j) = 2J M («r+|:(-i) B J M («r' 1 = VJ w (a)f (A) In precisely the same manner, if we consider the product of *W + [2] + ^L2][4] + • •• • + ^ {2w }. + '■■■ A [2j ; [2] +7 [2] [4] ( ^ {2rc}! + * Pvjc. J'.clin. Math. Soc, vol. xxii. THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. 117 we obtain El -(m) El -( J -^-') = Zp'^Mb^r+Zi- itf^mw- 1 )*-" v 2 \*y p'\ L^J ' »=o »=i = Jj> ntn -"%m(W- 1 )t n (/*> We have now, on taking the product of (X) and (m), +» +« / at \ / at~ i \ / bt \ / '» 2 "6^ _1 \ 2Jw<«)<" x Z^ < """ , Im(^'- 1 )^ = M[2])M"T2j) E ^([2]) E |.( ~ IXT ) The product of the four basic-exponential functions on the right of this expression is the product of two convergent series / , , (a + b)t, (a + b)(a + p*b)f 2 , I „ J n (a+P 2v b)t^ , (a + p»b){a + p»+* b) t^ ) , . i 1+ -[2]- + — T2]W rM 1 "— [2T~ + p^KT" ") () This result follows from result (>?) of article (9). If now we equate coefficients of the various powers of t in (B) with the corresponding coefficients in + 00 +00 remembering that J[«] = ( - 1 )"J[-«] 3 M = (-!)"![-»] we obtain from the terms which are independent of t JraHIra^jp"- 1 )-^ (p)< = -, _ (a + b)(a +p-"b) ~w~ which by an obvious reduction becomes Jto](«)3 [ o ] (^"- 1 ) -y^Jral")!^- 1 ) +..••+(- W^-'&JMMW- 1 ) - (cr> = J* (a,b) Equating the coefficients of t n we obtain m=+x Z J[».](«)|[»-».](^ , '" , )?' , "" ,, " , """ , ""=J;(fl.6) ■ ■ • (t> m=— oo the expression for JT being that given in article (8) expression (5). 11. When n is not a positive integer the expression (a + b)(a+2) 2 b) (a +p^~ 2 b) in J* (a , 6) must be replaced by J_j (a + &)(a+jr&)(a+jj 4 fr) .... (a+jfo-zft) „ «=»(a+y6)(a+_p 2 »+ 2 6) .... ( a +_p2n+at-2j) a TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 6). 19 118 THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. If. however, p > 1 , T ( a + bp^ia + bp 2 *-*) . . . (a + Sp 2 "- 8 ") ^ «ii (a +p- i b)(a + p~ i b) . . . {a+p- 2 «b) is the effective representative of the product (a + b)(a +p*b) .... to w factors. This corresponds to the change of n\ in the Bessel coefficients into T(n+1) in the case of Bessel-Functions. The series expansions of the products given above may be found in Proc. L.M.S., series 2, vol. i. pp. 63-88. The theorem analogous to Neumann's theorem J ( cl * + V + 2ab cos 8) = J (a)J (b) + 2^(-l) s J s (a) J. &) cos s6 . . (I) I have investigated in a paper (Proc. L.M.S.). The function E p being used in a manner similar to the use of the exponential (pp. 25, 26, 27, Gray and Matthew's Treatise on Bessel- Functions), gives us a rather complicated extension of (£). ( 119 ) VII. — On Some Points in the Early Development of Motor Nerve Trunks and Myotomes in Lepidosiren paradoxa (Fitz.). By J. Graham Kerr, Kegius Professor of Zoology in the University of Glasgow. (With Six Plates.) (Read January 18 ; MS. received February 9, 1904. Issued separately July 1, 1904.) CONTENTS. Introductory 119 Development op the Motor Nerve Trunks . 119 Development op the Myomeres . . . 122 PAGE General Remarks . . . . .125 Explanation of Plates . . . . 127 Introductory. My main purpose in the following short paper is to publish figures illustrating some of the more important facts of the early development of myotomes and motor nerves in Lepidosiren. The bearing of some of the observations of nerve development upon current theories renders it particularly desirable that they should be illustrated by untouched photographs of the sections. A few photographs illustrating the more important stages * in the development of the motor nerve trunks are given on Plate I. For the preparation of the photographs here published, as well as several others, I am indebted to the skill of my friend Dr T. H. Bryce, and of Mr Fingland, our University photographer. Development of the Motor Trunks of the Spinal Nerves. In describing the observed phenomena it will be convenient to begin with a late stage in development and work backwards to the earlier stages, and so pass from the better known and more familiar to the less known and less familiar. Stage 34. — Fig. 1 illustrates a considerable length of motor nerve from stage 34.t Here the nerve (n.t.) consists of a cylindrical mass of nerve fibres about 13/u. in diameter. On the surface of this the nuclei of the protoplasmic sheath are conspicuous. The greater part of the sheath itself is so thin as to be invisible even under the 3 mm. immersion objective, except in the neighbourhood of each nucleus, where it swells out to form a thick mass containing the nucleus. Stage 31. — At this stage the nerve rudiment on superficial examination presents the appearance simply of a chain of nuclei placed end to end in a strand of protoplasm. * By stage n I mean the stage represented by fig. n in my paper on the external features during development, Phil. Trans. Roy. Soc. B., vol. cxcii. p. 299. t Cf. Rapfaele's fig. in Anat. Anz., 1900, p. 340 (Per la genesi dei nervi da catene cellulari). Of. also Kolliker's remarks on this, op. cit., p. 511. TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 7). 20 120 PROFESSOR J. GRAHAM KERR ON SOME POINTS IN THE Appearances, in fact, support the Kettentheorie — suggesting a chain of " cells " placed end to end. More careful examination shows the presence within the protoplasmic strand of a cord, faintly fibrillated longitudinally, and differentiated from the simple proto- plasm by its affinity for eosin. The nerve runs downwards along the inner face of the myotome, and it is difficult to make out with certainty its connections with the cells of the myotome. Stage 29. — At this stage (figs. 2 and 7) the conducting part of the nerve (n.t.) is in a similar condition to that described for stage 30 — a distinctly fibrillated band — usually about 2 '5 to 3 m in thickness close to its root — which stains deeply with eosin. It slopes outwards and downwards from the ventrolateral angle of the spinal cord to the inner surface of the myotome, along which it proceeds in a ventral direction. The protoplasmic sheath (p.s.), however, is now far more conspicuous than in later stages. It is a great, irregularly thickened mass of granular protoplasm, sharply marked off from the true nerve by its being stained more greyish in colour in hematoxylin and eosin preparations, in sharp contrast to the deep red of the nerve trunk. Scattered through the protoplasm of the sheath are large nuclei rich in chromatin, yolk granules, and here and there vocuoles. Stage 27. — At this stage (figs. 3 and 8) the most conspicuous difference from stage 29 is that the nerve trunk (n.t.) is now naked for the greater part of its length. At its outer end it spreads out into a number of strands arranged in a conical fashion. In the case of the strands near the axis of the cone — i.e. in the case of the strands which pursue a direct course towards the inner surface of the myotome — it may be clearly seen under a high power of the microscope that each strand passes perfectly continuously and by insensible gradations into the granular protoplasm, which forms a tail-like process of a myoepithelial cell of the myotome. In the case of many of the motor trunks at this stage there is to be seen a mass of mesenchymatous protoplasm (p.s.) richly laden with yolk, and containing numerous nuclei, concentrated in the neighbourhood of the nerve towards its outer end. This is the rudiment of the mesenchymatous sheath which in stage 29 we saw had spread out over the surface of the nerve. The nerve trunk itself is about the same thickness as in stage 29, though I find considerable variation in this respect. Stage 25 (figs. 4 and 9). — A little behind the middle of the body at this stage the myotome is seen to be just commencing its recession from the spinal cord, mesenchyme (me.) cells * richly laden with yolk having begun to migrate in between the two structures. The nerve rudiment (n.t.) in the section figured is about 7m thick. It is distinctly fibrillated, and at its lower end expands into a cone as in stage 27 — the base of the cone, however, here being less expanded. Traces of longitudinal fibres are already visible on the ventrolateral surface of the spinal cord. * Here as elsewhere I use the word " cell " merely as a substitute for the more cumbrous expression " nucleated mass of protoplasm " without in the least implying that it is separate from its neighbours. As a matter of fact the " cells " of the mesenchyme are merely the enlarged and nucleated nodes of an irregular continuous protoplasmic spongework such as Sedgwick describes in Selachians. EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES. 121 Stage 24. — Transverse sections about the middle of the body at this stage (figs. 5 and 6) show the myotome still in contact with the wall of the neural canal, mesenchyme either not having intruded itself between the two at all, or only having done so to a very slight extent as yet except in the anterior part of the body. In embryos which have been excised while alive in normal salt solution, spread out in one plane like that figured to illustrate this stage in my account of the external features of development,* and fixed in this position, the myotomes are pulled slightly away from the neural tube. It is seen in transverse sections of such embryos that the rudiments of the motor nerve trunks already exist as soft thin bridges (n.t.) metamerically arranged and connecting spinal cord rudiment and myotome as shown in figs. 5 and 6. The nerve rudiments at this early stage are formed of granular protoplasm either without yolk or containing only very minute granules, without obvious fibrillar structure when stained by the ordinary methods. The rudiment is quite naked, the richly yolked mesenchymatous sheath of later stages being conspicuous by its absence. Nor are there any nuclei in contact with the nerve rudiment. There can happily be no doubt about this in Lepidosiren, where the nerve rudiment in this early condition is of relatively small size compared with the dimensions of a single nucleus ! The nerve rudiment — composed as it is of soft protoplasm — is at first extremely fragile, gradually becoming tougher as development proceeds. Consequently in a straightened out embryo of this period we find the more posterior nerve rudiments — i.e. those in an earlier stage of development— show a greater and greater tendency to be torn away from the myotome in prepared sections. A nerve rudiment which has become torn away from the myotome is shown in fig. 10, which brings out a point more difficult to observe in the uninjured condition, that the protoplasmic mass forming the nerve rudiment spreads out over the inner face of the myotome. How far this expansion extends, whether — as is probable — it really covers the whole face of the myotome, is a point almost impossible to decide definitely by actual observation. Similarly I am deterred by the unreliability of observations made on a spinal cord so laden with yolk in its early stages from saying anything as to the connections of nerve rudiments with neuroblasts or other cellular elements in the substance of the spinal cord. At this stage the spinal cord is without any obvious mantle of fibres. The motor nerve trunk has thus been traced back to a period in which it is repre- sented by a bridge of soft granular protoplasm connecting spinal cord and myotome at a stage when these structures are in close apposition. As the myotome becomes pushed outwards by the development of mesenchyme, it remains connected with the spinal cord by the ever-lengthening strand of nerve. As the nerve develops it soon loses its simple granular protoplasmic character and assumes a fibrillated appearance. Eichly yolked masses of mesenchymatous protoplasm become aggregated round the nerve, which till now has been quite naked. At first this protoplasm forms an irregular mass towards the outer end of the nerve trunk, but it soon spreads along it and forms a definite sheath. * Phil. Trans. B., vol. cxcii, pi. 10, fig. 24. 122 PROFESSOR J. GRAHAM KERR ON SOME POINTS IN THE This is at first very thick and conspicuous, but it gradually thins out, its yolk is con- sumed, and eventually the only parts remaining conspicuously visible are the nuclei dotted alon£ the surface of the nerve trunk. Into the subsequent history of the motor nerve — which is of minor morphological interest — I do not propose at this time to enter in detail. The protoplasmic sheath grows into the nerve trunk, dividing it up into separate bundles of fibrils. The nerve trunk, as has been shown, spreads out at an early stage in conical fashion over the inner face of the myotome. As the myotome grows in surface this cone becomes broken up into distinct strands which become more and more divergent. As the adult condition is reached the part of the nerve trunk proximal to the point of divergence — i.e. to the apex of the cone — increases relatively little in length. The distal portion, on the other hand, with its individual branches, increases enormously in length and the branches become more and more changed in direction as the muscles to which they are attached become pushed about by differential growth. Development of the Myomeres. The general features in the development of a typical myomere as seen in transverse sections are shown in text-figures A to H, # and in detail in PI. III. and IV. figs. 11-14. The protovertebra, at first (text-fig. A) a solid diverticulum of the enteric rudiment, develops a myocoelic cavity through the breaking down of its central cells about stage 21 (text-fig. B). By stage 24 (text-fig. C) the myotome is beginning to show signs of a flattening from without inwards. The myocele is obliterated, and the cells of its mesial wall have become flattened in form. At their outer ends they interdigitate with the inner ends of the cells of the outer wall, so that the line separating the two walls in a transverse section is a zigzag one. In stage 27 (text-fig. D, PL III. fig. 11) the inner wall cells have become more regular in shape ; forming rectangular parallelepipeds flattened in an obliquely dorso- ventral direction, so that their larger faces slope inwards and downwards. Contractile fibres (fig. 11, c.f.) have now appeared in the body of these cells, running longitudinally and forming most frequently a layer close to each of the dorsal and ventral surfaces, the two layers becoming frequently continuous with one another externally, and sometimes internally. Very often, however, the arrangement of fibres at their first appearance does not show this regular arrangement. The contractile fibres appear to be rounded in section, and are easily distinguished by their highly refringent character, and by their peculiarly deep stain with Heidenhain's hematoxylin. The cells of the myotome are at this stage still laden with yolk, and this naturally is a difficulty in the way of observation. The muscle fibrillse are striped almost from the beginning. One can often see in * These have been printed as separate Plates — V. and VI. EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES. 123 Heidenhain hematoxylin preparations that the young fibril appears to be made up of discrete particles arranged end to end.* But between these the protoplasmic matrix shows a more or less distinct fibrillar structure, the fibrils causing it having a longi- tudinal course and being continuous. These continuous fibrils appear to be an earlier stage of the contractile fibrils. The outer wall of the myotome in transverse section has the appearance of being composed of a single layer of large cubical epithelial cells. In horizontal sections the cell boundaries are less easy to make out, but when visible they show that the cells form a truly cubical epithelium. During mitosis these cells become more or less spherical. Stage 31. — At this stage (text-fig. E, and figs. 12 and 12a) the outer portion of the muscle cell has increased in size, so that the outer limit of the contractile fibres is- relatively much further removed from the outer end of the cell than it was before. In this outer end of the cell the cytoplasm assumes a clear transparent appearance, and in the preserved specimens large clear vacuoles are seen which possibly in the fresh condi- tion contained glycogen. The inner part of the cell is now almost filled with contractile fibres, the protoplasm being reduced to the matrix between them. In this matrix yolk granules are still abundant, and it is noteworthy that the muscle cells are now becoming multinucleate, the original nucleus having divided repeatedly. The division is mitotic. As the divisions only take place at relatively long intervals, a little patience is required in hunting for the mitotic figures. The resting nuclei lie free in the protoplasm of the myoblast. It is interesting, however, that during the period of mitotic activity the nucleus becomes surrounded by a sharply delimited more or less spherical mass of protoplasm, simulating the appearance of a cell within the myoblast. So striking is this appearance (PI. IV. fig. 14) that it suggested at first sight that more or fewer of the nuclei of the myoblast were really the nuclei of cells which had wandered into its substance from the mesenchyme without, just as such cells wander in later between the muscle fibres. On the whole, however, the balance of evidence is in favour of the cell-like structure round the nucleus being merely a temporary phenomenon due in some way to the influence of the mitotic activity of the nucleus on the surrounding cytoplasm — a phenomenon of the same nature as the rounding off into a spherical shape during mitosis of cells which in the resting condition are of more irregular outline. The appearance of the myotome of this stage, as shown in horizontal sections, is indicated in PL IV. fig. 12a. At about this period a striking change comes over the outer wall of the myotome. Numerous mitotic figures are observed in it. Its cells subdivide rapidly, so that the outer wall becomes several cells thick. The innermost of the cells so arising become squeezed in between the rounded ends of the primary muscle cells. At this stage it is often difficult to draw a line of demarcation between the outermost cells and those of the cutis which is now beginning to appear between myotome and skin. I am not, however, prepared to assert definitely that they actually give rise to cutis. * Cf. Godlewski, Arch. Mikr. Anat, Bd. lx., 1902. 124 PROFESSOR J. GRAHAM KERR ON SOME POINTS IN THE The outer wall cells now become converted into elongated irregularly cylindrical cells which stretch continuously from one muscle septum to the next. In the proto- plasm of these there begin to appear about stage 31+ (PI. IV. figs. 13 and 13a, and text-fig. F) contractile fibrils of a similar nature to those which have been long present in the inner wall cells. About stage 31 mesenchyme cells wander in between the myotomes. These give rise to the substance of the septum. Some also wander in between the muscle cells of the outer wall. By stage 34 the layer of muscle cylinders arising from the outer wall much exceeds the inner wall in thickness. It is distinguished from the latter at the first glance, its muscle cells being slender cylinders instead of flattened parallelepipeds. The general appearance in transverse section of a myotome of this stage will be gathered from text-fig. G. It will be seen that the lateral branch of the vagus nerve with its surrounding mesenchyme has formed an immovable obstacle, so that the myotome as it increases in thickness, and as it is pushed outwards by the increase of mesenchyme on its inner side, becomes gradually divided into two portions, a dorsal and a ventral, the two remaining for a time connected by a thin isthmus but being eventually completely separated. Along the mesial face of the myotome are seen the muscle cells of the inner wall, now reduced greatly in size in proportion to the whole thickness of the myotome. The whole thickness of the myotome outside this consists of the derivatives of the outer wall. In its extreme outer portion it consists of cylindrical cells still in the condition described for stage 31, in which contractile fibres are just beginning to appear. From this in a mesial direction the muscle cells are seen to become of greater and greater diameter, and their contained contractile substance increases, especially in size, showing that the muscle cells are older as they are farther removed from the outer surface. This, together with the fact that mitotic figures are numerous in the external layer, shows that this latter is the region in which growth in thickness of the myotome takes place. The myoepithelial cells of the inner wall remain distinct up to about stage 35, though constantly becoming more and more insignificant as compared with the great mass of the myotome composed of muscle cylinders derived from the outer wall. About the stage mentioned, however, the myoepithelial cells begin to break down, portions becoming segmented off from their outer ends. These resemble the muscle cells of the outer layer in character, being long cylinders. This process goes on, and soon the once conspicuous myoepithelial cells have become entirely resolved into these cylindrical elements, and the myotome is composed of apparently similar elements throughout its thickness (text-fig. H). EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES. 125 General Eemarks. 1. Motor Nerve Trunks. As is well known, there are three main views regarding the development of motor nerves in the vertebrata, which may be shortly stated as follows : — (1) Each nerve fibre develops as an independent outgrowth from a ganglion cell which gradually grows out towards, and finally and secondarily becomes united to, its special muscle. The sheath of protoplasm surrounding the nerve is an accessory structure of independent origin developed from mesenchyme. This view is associated especially with the name of His, and is the view favoured by the majority of embryologists. (2) The nerve trunk is multicellular in origin, consisting at first of a chain of cells, in the substance of which the nerve fibres are developed later, as fine fibres passing continuously from one cell body to another. The elements forming the original chain are most frequently looked on as ectodermal elements which have wandered out from the spinal cord rudiments. The protoplasmic sheath is derived from parts of the original cell chains which retain their protoplasmic character (Balfour, Goette, Beard, Dohrn, v. Wijhe, and others). (3) The nerve trunk is not a secondarily formed bridge between spinal cord and motor end organ. It has existed from the first, and in subsequent development it merely undergoes elaboration from its at first simple protoplasmic beginning (v. Baer, Hensen, Sedgwick, Furbringer and others). It is clear that the facts of development in Lepiclosiren, at least in the motor nerves of that animal, give strong support to the last-mentioned view as regards the nerve trunk itself, and to the second view as regards the protoplasmic sheath. It has been shown that by the examination of earlier and earlier stages the motor trunk can be traced back, without, I think, any possibility of error, to a simple protoplasmic bridge which already connects the substance of the medullary tube with that of the myotome at a stage when they are still in contact. As regards the origin of the protoplasmic sheath the evidence of Lepidosiren is equally emphatic. In its early stages the motor trunk is perfectly naked. About stage 27 masses of mesenchymatous protoplasm kden with yolk become applied to the nerve trunk, at first over only a small portion of its length, and these masses of protoplasm gradually spread over the whole trunk, remaining, however, for some time clearly distinguishable from the nerve trunk by their difference in staining reaction. As development goes on the yolk becomes used up, the protoplasm with its nuclei extends into the substance of the nerve trunk — doubtless to keep up the proper proportion between the bulk of the nerve trunk and its nutritive surface in contact with the sheath protoplasm. The protoplasm itself becomes less and less conspicuous, and eventually is only to be detected in the immediate vicinity of the nuclei. 126 PROFESSOR J. GRAHAM KERR ON SOME POINTS IN THE Lcpidosiren offers no evidence, so far, as to the ultimate origin of the nerve fibrils. They appear gradually in an at first simple protoplasmic matrix. They seem to form, as it were, a kind of picture in morphological symbols of already existent physiological activities. It would seem that the continued passage backwards and forwards between centre and end organ of a stream of nerve impulses gradually finds expression in the marking out of the original simple-looking granular protoplasm into definite fibrillar impulse tracks,* the undifferentiated protoplasm remaining as the perifibrillar substance. Such a view of the nature of the fibrils is supported by Bethe's remarkable observation t that in the chick embryo about the end of the sixth day a nerve trunk may lose its fibrillar character in the immediate neighbourhood of a mitotic figure, to reassume it on the completion of mitosis. It is pretty clear that the great function of the sheath is to serve as a nutritive organ for the nerve trunk. We see how its protoplasm is at first laden with yolk which gradually becomes used up as the nerve trunk develops within it. That the main function of nuclei, apart from reproduction, is to control cytoplasmic metabolism is well recognised. The nuclei of the sheath are able to exercise this control over the active metabolism of the developing nerve trunk which is without nuclei of its own. Connected with this relation of the sheath nuclei to the metabolism of the nerve trunk is no doubt the active multiplication of these nuclei observed in early stages of nerve regeneration.! In such regeneration it may well be that the protoplasmic matrix of the nerve simply repeats the process of its original develop- ment, increasing in size and then developing nerve fibrils within itself. If these fibrils represent merely the differentiated paths of nerve impulses passing through the substance of the protoplasm, it would of course happen naturally that the regenerated fibres would be formed in continuity with those of the undegenerated stumps. On this view the process which takes place when the peripheral part of a cut nerve degenerates and then regenerates is somewhat as follows : — The fibrils, no longer subject to the stimulus of passing nerve impulses, revert to their proto- plasmic condition. The protoplasmic sheath becomes highly active. § It increases in thickness : its nuclei divide actively. Its protoplasm digests the remains of the medullary sheath. || It thus comes to contain stored- up food material as in its original embryonic heavily yolked condition. The protoplasmic matrix representing the degenerated axis cylinder lies imbedded within the sheath.ir Controlled and supplied with nourishment by the activities of the surrounding sheath the protoplasm behaves just as it does in ontogenetic development: (l) it grows — probably slowly — and * Were this the case, it might well be that the formation of fibrils might tend as 1 a rule to spread from the end of the nerve from which came the most active and frequent nerve impulses. t Bkthe, Allgemeine Anatomie unci Physiologie des Nervensystems, p. 244. X Bungner, Ziegler's Beitrage z. Path. Anat., Bd. x., 1891. § Wieting, op. cit., Bd. xxiii., 1898. | This view of nerve regeneration, which my ontogenetic work inclines me towards, appears to agree most closely with that enunciated by Neumann {Arch. Path. Anat. u. Phys., Bd. clviii. p. 466). IT This protoplasmic strand within the protoplasmic sheath could only be demonstrated with extreme difficulty. EAELY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES 127 so gaps are bridged over ; and (2) as soon as it has become continuous, nerve impulses beginning to play backwards and forwards in its substance cause again a differentiation into fibrils. As part of the impulse tracks persist as the stumps of the fibrils, the regenerate parts of the fibrils will naturally develop in exact continuity with these. I have no intention of entering into a general discussion either as to the data in connection with the development of nerves in other Vertebrates, or as to the general conclusions which have been based on them. My purpose now is merely to emphasize the observations of the phenomena as they occur in Lepidosiren. I may, however, be allowed to point out that such observations as those of Bethe upon the Chick, * although apparently supporting the cell chain view, are in no way irreconcilable with the observations here chronicled. The soft protoplasmic bridges which form the first distinguishable rudiment of the motor nerve are difficult to observe in Lepidosiren, whose histological features are upon a relatively large and coarse scale ; and how much more delicate and difficult to observe are the corresponding structures likely to be in the chick ! It may well be that further research will demonstrate the existence of a delicate protoplasmic bridge about which are clustered the " nerve cells " (in Apathy's sense) observed by upholders of the cell chain theory. DESCRIPTION OF PLATES. General List of Abbreviations. c./. Contractile fibrils. nu. Nucleus. d.r. Dorsal root. n.c. Neural canal. g. Glycogen-containing outer portion of myoblast. n.s. Nuclei of protoplasmic sheath. i.w. Inner wall of myotome. n.t. Nerve trunk. l.vag. Lateral branch of vagus. o.w. Outer wall of myotome. me. Mesenchyme. p.s. Protoplasmic sheath of nerve trunk. m.n.r. Motor nerve root. s.c. Spinal cord. my. Myotome. v.r. Ventral root. n. Notochord. y. Yolk granules. Plate I. [The figures on this plate are all from untouched negatives.!] Fig. 1. Portion of spinal nerve of stage 34. (108'28 35.) l.vag., lateralis vagi; my. myotome; n. notochord ; n.t. nerve trunk ; n.s. nuclei of protoplasmic sheath ; d.r. dorsal root ; v.r. ventral root. Fig. 2. Portion of spinal nerve trunk of stage 29.* (93 C. 1435.) my. myotome; n. notochord; n.t. motor nerve trunk ; p.s. protoplasmic sheath containing yolk granules (black), and large nuclei rich in chromatin. * Op. cit., p. 238. t Much of the minute detail has unfortunately disappeared in the mechanical printing of these figures. I shall be glad therefore to show any specialists who are interested sun prints from the same negatives, in which the full detail is brought out. TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 7) 21 128 EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES. Fig. 3. Part of transverse section at stage 27. (69 C. 1064.) Showing whole length of motor nerve trunk, my. myotome ; n. notochord ; n.t. motor nerve trunk ; p.s. protoplasmic sheath now in form of a mass of richly yolked mesenchymatous protoplasm aggregated round the nerve trunk near its outer end. The greater part of the nerve trunk is still naked. At its outer end the nerve trunk passes out into conically arranged strands of protoplasm forming the inner ends of the muscle cells. Fig. 4. Part of transverse section at stage 25. (79 A. 933.) my. myotome ; n. notochord ; n.t. motor nerve trunk already faintly fibrillated passing between spinal cord (s.c.) and myotome. The mesenchyme (me.) has not yet begun to concentrate round the nerve trunk. Fig. 5. Part of transverse section at stage 24. (73 G. 986.) my. myotome; n. notochord; n.t. motor nerve trunk, at this time composed of granular protoplasm, and naked. Fig. 6. Similar section to last, but taken from rather less advanced specimen. (Stage 24 ; 73 F. 688.) my. myotome ; n.t. motor nerve trunk — a naked protoplasmic bridge connecting myotome and spinal cord (s.c). Plate II. [All figures are camera drawings of single sections. Figs. 7, 8 and 9 represent the same sections as are photographed in figs. 2, 3 and 4, with the additional detail visible under Zeiss' 3 mm. apochromatic homogeneous immersion objective.] Fig, 7 ( = fig. 2). The motor nerve trunk (n.t.) is seen within its thick protoplasmic sheath (p.s.). Fig. 8 ( = fig. 3). The motor trunk is now naked except for the large mass of yolk-laden mesenchymatous protoplasm (p.s.) which has concentrated round it towards its outer end. The continuity of nerve trunk with protoplasm of muscle cell is seen. Plate III. Fig. 9 ( = fig. 4). The nerve trunk is seen to be already fibrillar in structure. Mesenchyme (me.) has penetrated in between myotome and spinal cord, but has not yet begun to concentrate round the nerve trunk to form its sheath. Fig. 10. Stage 24. The nerve has been torn away from the inner surface of the myotome so that its expanded outer end is seen. Fig. 11. Part of transverse section, stage 30. (93 B. 2844.) The two walls of the myotome are seen — the outer one-layered, the inner composed of a layer of myoblasts in which are seen the first contractile fibrils (c.f.). Fig. 12. Part of transverse section, stage 31. (103 D. 2463.) The outer wall of the myotome is beginning to show more than one layer of nuclei. In the myoblasts of the inner wall the contractile fibrils (c.f.) have greatly increased in number, most of the protoplasm of the inner half of the cell being converted into fibrils. Plate IV. Fig. 12a. Part of longitudinal horizontal section at stage 31 (103 C. 822), showing a single myotome. i.w. Inner wall cell ; c.f. contractile fibrils ; g. glycogen-containing outer part of cell ; nu. nuclei of inner wall cell ; y. yolk granules ; o.w. outer wall of myotome, two of the nuclei undergoing mitosis : me. mesenchyme nuclei of septum between myotomes. Fig. 13. Part of transverse section at stage 31 + . (106 C. 1573.) The outer wall of the myotome is now several layers thick, and the cells of this wall have also developed contractile fibrils (c.f,). Fig. 13a. Longitudinal horizontal section through a myotome of stage 31 + . (106 A. 1811.) The nuclei of the inner wall myoblast are seen to have considerably increased in numbers. The contractile fibres of the outer wall (c.f.) are visible. Fig. 14. A nucleus of one of the inner wall myoblasts during mitosis, showing the cell-like demarcation of the protoplasm immediately surrounding it. v r: Early Development of Motor Nerve Trunks and Myotomes in Lepidosiren paradoxa.— Plate I. Sol. Edin. my FIG. I. FIG 4- FIG 2 FIG 5. Wm { \ / trunk. I protopla re II. -fk on i on ions. Figs. 7 ; e same sections as are 3 mm. apochromatic Co ^ ^ ^?°° f^^/p A* Q O o I e an s\/' X' ( . . <tf) the nreQ^niV^s oeaiiinJAur - nifccontractile fibrils\|(c v S OH „ lew-Turns' iiuc •'P^iwng mitf • '■ ; ng a single myotome. c on i r: Early Development of Motor Nerve Trunks and Myotomes in Lepidosiren paradoxa.— Plate I. ySoc. Edin. Vol. XLI. B00 hB& ■s Vans. Roy. Soc.Edin 1 KERR^ EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES IN LEPIDO SIREN PARADOXA. PLATE 2. Vol.XLI Fig. 7. Pig.8. E .Wilson , Cambridg e . Roy. Soc.Edm T KERR: EARLY DEVELOPMENT OF MOTOR NERVE TRUNKS AND MYOTOMES IN LEP1DOSIREN PARADOXA. PLAT XL I. E Wilson, Cambridge y- Soc.Edm 1 Vol.XLI. KERR: EARLY DEVELOPMENT OE MOTOR NERVE TRUN] AND MYOTOMES IN LEPIDO SIREN RARADOXA . PLATE 4. Figs. 12 A, 13, 13A. I I I I I I I I I I I I c.f! o.w. 13A. ■~.yv E . Wilson, Canibridg e . Trans. Roy. Soc. Edin. Vol. XLI. Prof. J. Graham Kerr on "Some points in the Early Development of Motor Nerve Trunks and Myotomes in Lepidosiren paradoxa." — Plate V. DESCRIPTION OF TEXT FIGURES A-H. Camera outlines of portions of transverse sections of the trunk region of young Lepidosirens at various stages, to show the topographical relations of the myotomes. Zeiss Obj. A, Oc. 2. i.w., Myoblasts of inner wall of myotome ; /. vag., lateral branch of vagus. Fig. A. — Stage 16. Fig. a— Stage 21. Fig. C. — Stage 24. Fig. D. — Stage 27. Fig. E. — Stage 31. Fig. ¥. — Stage 31 + Trans. Roy. Soc. Edin. Vol. XLI. Prof. J. Graham Kerr on "Some points in the Early Development of Motor Nerve Trunks and Myotomes in Lepidosiren paradoxa."— Plate VI. 1 ' £glr* •• : - ft ■ W°C - - •* .. „"•> ZZ-SjfU fj f, -sv I Fig. G.— Stage 34. ;... Pig. H. — Stage 36 + . ( 129 ) VIII. — An Analytical Theory of the Equilibrium of an Isotropic Elastic Plate. By John Dougall, M.A. Communicated by Professor G. A. Gibson. (MS. received February 4, 1904, Read March 21, 1904, Issued separately August 5, 1904.) The following paper contains a purely analytical discussion of the problem of the deformation of an isotropic elastic plate under given forces. The problem is an unusually interesting one. It was the first to be attacked (by Lame and Clapeyron in 1828) after the establishment of the general equations by Navier. The solution of the problem of normal traction given by these authors, when reduced to its simplest form, involves double integrals of simple harmonic functions of the coordinates. The integrals are of complicated form, and practically impossible to interpret, a fact which, without doubt, has had much to do with the neglect of the problem in later times, and the almost com- plete absence of attempts to establish the approximate theory on the basis of an exact solution. An even more serious defect of Lame and Clapeyron's solution is that the integrals, as they stand, do not converge. A flaw of this sort has often been treated lightly by physical writers, the non-convergence of an integral being regarded as due to the inclusion of an infinite but unimportant constant. In the present case, however, the infinite terms are not constant, but functions of the coordinates, and the modifica- tions necessary to secure convergence, so far from being unimportant, lead directly to the most significant terms of the solution. The next writer to deal with the exact problem was Sir W. Thomson, who, at the end of the memoir in which he solved the problem of a spherical shell, indicated the form which the solution would take in the limiting case of a plate. His method, if carried out, would lead to integrals of the same form as Lame's. Solutions of special problems have been given by other writers. Prof. Lamb has worked out the solution for an infinite solid subjected to normal pressure proportional to cos kx, and verified in this particular case some of the results of the approximate theory of thin plates (Proc. Lond. Math. Soc, vol. xxi., 1889-90). The history of the approximate theory is well known and easily accessible. It will be sufficient here to refer to — (i) Todhunter and Pearson's History of the Elasticity and Strength of Materials. (ii) Clebsch's standard treatise, Theorie de Velasticite des corps solides, as trans- lated by Saint Venant.; in particular, Part I. chap, iii., and Saint Venant's brilliant note on § 73. (iii) Prof. Love's Treatise on the Theory of Elasticity, 1892, — especially the historical introductions to both volumes. The various forms of the approximate theory rest partly upon the general equations of equilibrium, partly upon auxiliary hypotheses or physical principles. These principles are recognised as contained in the general equations, but on account of the TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 22 130 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF analytical difficulties in the way of deducing them rigorously, they are either simply assumed, or else supported by reasoning plausible rather than demonstrative. In the following pages the problem is treated as a purely mathematical one, and the approximate theory for a finite plate deduced from an exact. solution for an infinite plate. The main features of the method are — (i) The use of Bessel functions in place of the simple harmonic functions of previous writers. Only the symmetrical forms, or functions of order zero, are required. (ii) Transformation of the definite integrals, in terms of which the solutions are in the first place obtained, into series, by means of Cauchy's theory of contour integration and residues. The series involve Bessel functions of the second kind with complex argu- ment, and are so highly convergent that the principal features of the strain represented by the solution can be made out with the utmost ease. (The transformations belong to a class discussed systematically, apparently for the first time, in a paper " On the Determination of Green's Function by means of Cylindrical and Spherical Harmonics," Proc. Eclin. Math. Soc, vol. xviii.) (iii) Detailed solutions of the problems of internal force with vanishing face traction. The usual method of dealing with a general problem in Elasticity is to find a particular solution for the bodily force, and then to treat the problem of surface tractions completely. This is theoretically sufficient, but leaves the result in a complicated form, which in the present case must be simplified before practical applications can be made. (iv) Use of Betti's Theorem (Love, Elasticity, vol. i. § 140) to develop a method analogous to the method of Green's function in the Theory of the Potential, by which the properties of the solution for a finite plate can be deduced from the infinite plate solu- tion. (Cf. Proc. Eclin. Math. Soc, vol. xvi., "On a general Method of Solving the Equations of Elasticity.") The results of the ordinary theory are fully confirmed, and extended in various direc- tions. The infinite solid solution gives, of course, an exact particular solution for internal force and traction on the plane faces of a finite plate. At the head of the solu- tion appear the terms given by the approximate theory. In the case of flexure, the equation of Lagrange is obtained to a second approximation. The problem of a finite plate under given edge tractions cannot be completely solved, but exact solutions are given of certain problems relating to a circular plate. For a thin plate, with edge of any shape, the conditions satisfied at the edge by the principal terms of the exact solution are found to a degree of approximation beyond the reach of any theory which rests merely on the " principle of the elastic equivalence of statically equi- pollent loads." For example, the celebrated boundary conditions given by Kiruhhoff, in correction of Poisson, are verified, and extended by the inclusion of terms of higher order. In conclusion, it may be mentioned that the methods given here are equally appli- cable to the problem of the vibrations of a plate, and to the problems of the equilibrium and vibration of a finite circular cylinder, or of an open spherical shell. Some account of these applications I hope to publish shortly. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 131 TABLE OF CONTEN T S. (i), 0)- <*)• PAGE Introductory Analysis 132 (a) to (</). Evaluation of definite integrals involving the Bessel functions. Degenerate cases . 132 (h) Potential functions derived by integration from the point-source potential. Definite integral expressions for these 135 Solution of the problem of flow between two infinite parallel planes. Main stream and local currents 136 Curvature. Differentiation as to arc and normal ....... 139 8 1. Equations of equilibrium. Form of solution for a plate free from bodily force . . 140 § 2. Force applied at a single point of an infinite solid 142 § 3. Solution of the problem of normal traction for an infinite plate ..... 143 § 4. Flexural and extensional components of the strain. Disadvantages of the solution in definite integrals ..... 145 § 5. Transformation of the definite integrals into series, by means of Cauchy's Theorem of contour integration 147 § 6. Types of the particular solutions composing the general solution 1 48 § 7. Position of the zeroes of the functions sinh C±C 150 § 8. Approximate forms of the »t th terms of the infinite series, when n is large . . . 151 § 9. The solution for arbitrary normal traction. Questions for discussion . . . . 151 § ] 0. Detailed solution of a special case. Term by term differentiations . . . . .152 § 11. The same special problem. Summation of two infinite series 154 Final form of the same special solution . . 155 Order of magnitude of the various parts of the solution when the thickness of the plate is small . . . . . .156 Methods and results of the sjJecial case ex- tended to the general problem of arbitrary normal traction 157 § 15. Independent symbolical solution of the general problem ...... 160 § 16. The problem of tangential face traction. Solution for an element of traction . . 161 § 17. Composition of the solution . . . .163 § 18. General solution. Comparison with the solution for normal traction . . .165 § 19. Normal force applied at a single internal point. Solution in definite integrals . 1-65 § 20. The same solution in series . ... . 169 § 12. §13. §14. § 21. Solution of a special problem of internal areal normal force ..... § 22. Approximate forms of displacements and stresses in the general case .... § 23. Normal force of constant intensity throughout the thickness ...... § 24. Internal force parallel to the faces § 25. Solution ....... § 26. Approximate results. Lagrange's equation for flexure to a second approximation § 27. Extensional strain. Differential equations of the principal mode ..... § 28. Approximate values of the stresses across a plane parallel to the faces .... § 29. Transmission of force to a distance. Expan- sions in polar coordinates .... § 30. Types of deformation conveying a given resultant stress Conditions for the existence of a solution with finite potential energy. Elastic equivalence of statically equipollent loads Betti's reciprocal theorem. Verification of preceding solutions Finite plate under edge tractions. Form of the solution deduced by means of Betti's Theorem The same by another method General solution for an infinite solid under any forces ....... § 36. Betti's Theorem and the problem of given edge tractions § 37. Exact solutions of special problems for a circular plate. Problem 1 — symmetrical transverse displacement .... § 38. Problem 2 — normal displacement and nor- mal shearing stress given. The Fourier and other methods of obtaining such solutions ....... §39. Problem 3 — permanent modes due to sym- metrical edge tractions .... § 40. Expansions of arbitrary functions . §41. The problem of given edge tractions for a thin plate ....... § 42. Extinsioial strain ..... § 43. The Green's function method for the per- manent mode §§ 44, 45. Flexural strain. Solutions to first and second approximation . . . 218, § 46. Flexural strain. The Green's function method. Kirchhoff's boundary conditions to a second approximation Addition to Paper §31. §32. §33. §34. §35. PAGE 170 173 175 176 178 180 181 182 183 185 187 189 192 195 196 197 198 201 205 206 208 208 213 220 224 228 1 32 y\\\ JOHN DOUGALL ON AN ANALYTICAL THEORY OF Introductory Analysis. (a) The Bessel function J is defined by the series Z m / Z 2 2 4 JmOO = 2 m nm\ ~ 2 • 2m + 2 + 2 • 4 • 2m + 2 ■ 2m + 4 For the function of the second kind we take as definition 2 sin m-n- This makes G-,„z an analytic function of m , the value of which, when m is a real integer, is G m ( Z ) = (log2- r + |).J m ( 2 )-Y m (,) where Y m (z) is Neumann's function. In this case, therefore, G m z = — log zJ TO z + a uniform function of z. In the following pages we are concerned chiefly with the function of order zero ^-r+T^-p+KP )-iog*(i-|J + ^ 2 - ■ ■ . )-£j + (i+^L- . When mod z is very large, while the phase (argument) of z lies between — ir/2 and Stt/2, then approximately G m z = e —e<>+4 } 2z Similarly, when the phase of z is between and 7r (excluding those values) (6) If .'• , y , z and p , a> , z are the rectangular and cylindrical coordinates of a point in space, so that x = p cos w , ?/ = p sin w , then the most important property of the Bessel Functions is that each of the eight functions (e KZ or e~ KZ ) (J m Kp or G m Kp) (cos wito or sin mot) satisfies Laplace's equation, or in other words is a potential function. Hence (y 2 + k 2 ) • (J m /cp or G m Kp) (cos mu> or sin mm) = . Further, if = p* + p * — Ipp COS (OJ - CD ) then (V 2 + *'-') • (J «R or G kR) = . Let now I = / / G kK/(:c', y')dx'dy\ the integral being taken over a finite area A. Then ( v 2 + « 2 )I = , if (as , y) is without, THE EQUILIBRIUM OP AN ISOTROPIC ELASTIC PLATE. 133 but ( v 2 + k 2 )I = - 27r/(a; , y) , if (x,y) is within this area; as easily follows from the theorem V 2 f f log Rf(x', y')dxd,f = 27rf(x , y) . The differential equation satisfied by I, together with the conditions that I and its first derivatives dl/dx , dljdy are continuous throughout, define the value of the integral completely, and in many cases make its evaluation easy. (c) For example, takef(x , y) = J m /3p cos m&> , with m an integer, so that I = l I G kR J m f3p' cos mm' p dp dm and suppose the area of integration to be a circle of radius a, with centre at the origin. For convenience in the proof, let the imaginary part of~/c be positive. Then I = -^ J m /3p cos m<o + AJ m <p cos mm , when p<a , p z — k- = BG„,Kp cos mm , , when p>a A , B are determined from the conditions that I and dl/dp are continuous at p = a. Thus we find I = „ 2 ~^" 2 J m /3p cos m <» + 02 ^ 2 J "' fcp cos mm ( Ka ^ r m xaJ m /3a - G m Kaf3aJ J fta) ; (p< a) = n 2 _ 2 ^m K P cos mw(KaJ m 'KaJ m /3a - J m Ka^aJ„'fSa) ; (p> a ) By the principle of continuation in the Theory of Functions, the result is true whatever be the phase of k. But when the phase of k is diminished by 2n, G m (/<c) is increased by 27riJ m (i<c) and G m '(Kc) by 27rz'J m '(Kc) ; hence, equating the corresponding changes in I and its value, we obtain I I J /cR J m Bp cos mid p dp dm J J 2tt = 02 _ 2 ^m K P cos m(ji(KaJ m 'KaJ m /3a ~ J„,Ka^aJ m 'fSa) . From this again it easily follows that in I and its value we may replace the G functions by the Y functions. (d) We have Y /cp = log KpJ Kp + £* 2 p 2 = log kJ k P + log p(l - |kV . . . ) + |kV . . . Thus log/c J Kp - Y Kp is an integral function of k, in which coefficient of k° is — log p , and coefficient of k 2 is ^p 2 log p - \p 2 . 134 .MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF The functions log p and ^p 2 \ogp — ^p 2 are thus in a sense degenerate forms of the Bessel Functions, and any theorem relating to the G or Y functions will yield a corresponding theorem in these. Thus by equating coefficients of k 2 in the equation ( v 2 + « 2 )(logKJ K /3 -Y K P ) = we obtain V 2 logp = V 2 (V log p-{p 2 ) = log p and therefore V *(ip 2 logp-ip 2 ) = 0. We deduce at once V 2 jf(W- log R - \W)f(x, y')dx'dy' = jf log Ef(x, y')dx'dy' and v4 // (iR2 l0g R " * R2)/(a; '' y') dx ' d y' = v '// lo § M<y')dx'dy' = 2tt/(* , y) . (e) Again, from the addition theorem Y /cR = Y KpJ Kp'+ 2 y ,Y m KpJ m Kp cos m(u - oj") ; (p>p) m=l we deduce logR = logp - 2,— f -) cos m(o> - a)') ; p>p' tri"Ap/ and iR2 log R - JR* = (Jp2 log p - |p2) + ip'2 log p + | (P - 2p log p)-^- - |_ | cos (oi - o>') ( /') In the same way, from the results of (o), we may deduce the value of the integral ^o = I / (4R 2 1°§ -K ~ iR 2 )Jm/3p' cos moi p dp dm' . The form of the result varies in the cases m = 0, m=l , m>l. m=0 : Io = ^ofr + J,{ {^ ~ P j)(loga/3aJ '/3a- J /3a) + |(l- ] og a\l3aJ '(3a + ?- f 2 log a - 1 JJ /3a 1 , p< a 7n=l : $ { ( 4 ^ l0§ P ~ 4 p2 )( ~ ^ aJ o'^ a ) + log p ( 7p - -J^o'/ 30 + \ J oP a ) } . P > a • + ?Vl + 2 log a) J^a + 2?(l - 2 log aWr/jSa \ , P <a .J-ac*. j^£ + l p2 _ l^iogp^a-^aJ^a) + |'(/taJ/j8a - SJ^a) } , p>a. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 135 m >1 : 2tt t n 2tt p m COS mm ( ( 1 p 2 Yo t'/d , T o^ I = _J m /3pcosma, - _L^_| (^ - j-E^lfaJ.pa + rnJ^ + 7 — ^2\ftaJ m 'f3a + m - 2 J m /3a ) > , p<a The corresponding integrals with log R in place of ^R 2 log R - ^R 2 may be obtained at once by taking y 2 0I> * ne above. Also, all through we may write log (R/c), log (p/c), log (a/c), instead of log R, log p, log a, this amounting merely to a change in the unit of length. (g) By equating coefficients of like powers of /3 in the results of (c) , (/) we can obtain I I G /cRp' m+2 "cos moi'p'dp'dm J o J o and T P^R 2 log R - iR 2 )p""+ 2 » cos mm'p'dp'da. In the case when to = n = // Q a KRp'dp'dw' = — — J Kp«aG 'Ka , p <a y K 2 K J = — — „ G Kp/<aJ 'Ka , p>a . Il llX \ R2 l0g R ~ I B ^)p' d P d<0 ' = § { P * + 4p2a2(2 lo S " ~ J ) + " 4 ( 4 !og « - 5) 1 , p < a = ( jP 2 logP - -p 2 J7ra 2 + logp^ , P >a. These results and those of (f) may easily be verified, or obtained, from the values dl d of v 4 of the integral, with the conditions that I , -r- , y 2 I > ^TV 2 ! are continuous at p — a. (h) In certain problems a class of potential functions occurs, which may be deduced from the fundamental potential l/r, where r 2 — x 2 + y 2 + z 2 , by successive integration with respect to z. Writing u x = log (r + z) u 2 = z log (r + z) - r u s = i( z2 ~ IP 2 ) lo g (r + z)~ \rz + £p 2 we may easily verify that u lt u 2 , u s are potential functions, and that du s ~dz = u., du.y dz = «i du x 1 ~dz r 136 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF These ^-integrals of 1/r may be expressed in the form of definite integrals involving the Bessel function J, analogous to the integral forms for r~ l and its z-derivatives, r d •~ l = i e-" z J Kpc?K f Y r~ l = I ( - K)e ~ KZ J KpdK , etc., where z > , ran We may notice that the value of I e'^^pch follows at once from the remark that it is a potential symmetrical about the axis of z, and taking on that axis the value f° 1 I e~ KZ (h = - . We may use this idea to express w : , u 2 , u 3 in similar form. z For we have j"( e - K z_ e - Ke yjL = -log-*, and /; \dK . z q-kz _ i + KZ e- KC — ^ = zlog z J K Z ° C 1 „ „ \d K 1 „. 2 3 f a (e-*-l + KZ - ^¥r«^= - J Z 2 1og^+ ~z- by integration with respect to z from to z. Hence r + z = - l0 §'2c I (e-" z J Kp- l + Kze- Ke j- (Ik , r + z because in each case the functions equated are symmetrical potentials, taking the same value on the axis of symmetry. By putting z = in the first and third of these we obtain two integrals, of great importance in the following analysis, f o (j « P - e -«)^= -io g -£ There is no difficulty in generalising the above results, but those given are all that we shall require. (i) With a view to indicating the broad lines of the treatment ' of the elastic problem given in the succeeding pages, a discussion on similar lines may be given here of a simple problem in potential, in which the attention is not distracted from the principles of the method by any complexity in the calculations. The problem is to find the flow from a source situated between two parallel planes z = db h, under the condition that there is no flow across these planes. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 137 We require a potential V, becoming infinite as l/r &t (x' , y f , z'), but with no dV ■ ' other singularity at a finite distance, and such that -77 = when z = dzh. then r~ l = / n~ K<z ~ z '\J KRdK , when z>z J = I e K ' ^ ~ ^ ' , J^HdK , when z<z. J u — r~~' = I ( -K)e~ K{ '-"J K'RdK, when 2>x' = I Ke" ,,,- * ,, .T icKd/c , when g<z'. I .' We therefore begin by finding a potential Y x = (A cosh kz + B sinh kz)J kR giving We obtain - d j- = Ke- K|2 - Z ''J KR on z = 7i = - k^ u - j, 'J a kR on z= -h. A sinh k/< 4- B cosh k// = e^ Kl ' i ~ z ' > A sinh k/i - B cosh nk = e~ K(h+x '' cosh kz A = e-" A B = e~ Kh sinh k7' sinh kz cosh k/> v - -<cj/ cosn k;2 cos h *z' sinh kz sinh kz'\ t y \~ e [ I-rCTS + TTTi— I J«V V sinh kIi cosh /c/t cR If this could be integrated with respect to k from to 00 , we should have a potential just balancing at the boundary the flow from the source. But V x becomes infinite as l/ich at /c = , and the integration cannot be performed. We may, however, subtract from Vj the (constant) potential e' KC JKli , where c is an arbitrary positive quantity. This makes integration possible, without introducing any flow across the boundary. A solution of the problem is then L f[ , /cosh kz cosh kz' sinli k2 sinh kz'\ - „ e~ KC V = - + /"°° { , /cosh kz cosh kz sinh k2 sinh kz\ „ e~ KC | J e -Khl ^_^ + \j kR _ V , J „ ( \ sinh k/i cosli /c/i y ° K/i ) But this form of solution, while theoretically complete, is of little value because of the difficulty of interpretation. For example, it gives no indication of what on physical grounds we should expect to be the chief feature of the phenomenon, namely, the practically two-dimensional character of the flow at a moderate distance from the source. The transformation to which we proceed brings this out as luminously as possible. First, it is convenient to separate Y 1 into its odd and even parts in *, as is easily done by writing cosh k/i — sinh <h for e~ Kh . TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 23 138 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF Thus / cosh ah ■> i > sinh kIi ■ , • 1 At -o ^i = ( - cosh kz-z' H cosh kz cosh kz - sinh kz sin Iikz )J kR . 1 \ sinh kH cosh k/i / Next we replace the term 1/r in V by the equivalent integral /" e^'-^JoKEdK . Hence V = / i s ; n h kz-z 4- K cosh kz cosh kz' — sinh kz sinh kz' )J kR - — — > i Jo (V sinh x/i cosli k/i J kIi ) the upper or lower sign being taken in the ambiguous term according as z is > or < z'. When R > , this integral can be separated into the two /'% ,../ , cosh x/i , , , sinh k/i . , . , , 1 \, J„kR + sinh kz - z + ^-j — 7 cosh kz cosh kz — - sinh kz sinh kz - Wk J u \ sinh K/i cosh k/i khz 1 R The value of the latter inteoral we have found to be — -=- log tt • The former integral is of the form I J,/RF(/c)<i/c, where F(*r) is an odd function of *c, vanishing for < = 0. It may be expressed as a complex integral — ; I G kRF(k)cZk, the path being from west to east along the whole of the real axis in the k plane, for G (kR) - G (k^R) = 7rt'J KR . Now, from the original form of V 1? and the integral forms of l/r, it is obvious that F(/c) vanishes at infinity in the eastern half of the k plane ; being odd in k it must vanish likewise in the western half. Hence by Cauchy's Theorem, the integral — ./G kRF(>c)o?/c is equal to twice the sum of the residues of the function G kRF(/c) at its poles in the upper half of the < plane, and v = - Nl 2 "^ n-irz nirz inirK V — jL^ cos — j— cos . - G II , l = l /i li " /i 1 Y'wB 2/ A + lZ siu ( n + lTi sm ( n + lTi G ( n + (j) The solution indicates (i) a main current in two dimensions, defined by the 1 h ■I T> potential — j log 9 , and (ii) an infinite series of local currents in three dimensions, THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 139 practically insensible when the distance from the source is a moderate multiple of the thickness of the plate. In the following pages we shall deduce analogous solutions for ' sources of strain ' of the different types which may exist in an elastic solid, and develop these solutions in various directions. The corresponding development of the present solution is extremely easy, but would carry us too far. We merely mention that the ' main current ' in the hydrodynamical problems corresponds to the ' principal modes of strain,' the determination of which is the object of the theory of thin plates. But there is one important distinction in the two cases. In the flow problems the exact conditions defining the ' main current ' can always be found, and are indeed obvious ; on the other hand, the analogous conditions in the strain problems can only be found by approximation. (k) The following conventions seem to be very generally adopted, but to prevent any risk of ambiguity they may be stated explicitly here. Consider any continuous plane area A bounded externally by a closed curve C , and internally by one or more closed curves Cj , C 2 , etc. At any point E of a bounding curve let Ex , E// be drawn in the directions of the rectangular axes of coordinates. Let Ex , Ei/ be turned through an angle e, which will be taken as positive when the rotation is counter-clockwise, until they coincide with E£ , E>? , the direction of E£ being that of the normal at E when drawn from within A towards the boundary. E£ , E>; will be taken to be the positive directions of the normal and tangent at E, and \£f(x, y) be any function given within A, -/ and ~ will be used to denote the rates of variation of f per unit length in these CL7Z (a/o positive directions. The curvature at E is -^ and is denoted by I /p. p is therefore positive when, in order to reach the centre of curvature from E, we have to proceed into the area A. If we suppose the figure traced on level ground, a person proceeding along the boundary in the positive direction will have the area on his left, and the curvature will be positive when he is rotating about the vertical in the counter-clockwise sense. The following formulae relating to differentiation along the arc and normal will be much used in the later sections of the paper. Suppose the axes of x and y to coincide with the positive normal and tangent at a point of the bounding curve. At a neighbouring point E (x, y) on this curve df df , . dr -f- = COS e — + Sin e -z- , dn dx dy ' df ■ df , df I -j- = — sin e ~ + cos e — [ ds dx dy , By putting x, y, e equal to zero, we have at df = df 4f = df dn dx; ds dy ^ ' 140 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF Differentiate the first of equations (i) with respect to -s. Thus ( d £) = C0S€ < L( < V) \dnj ~ ' ds \dx) d_ (df ds d (df\ . <h df de df + sin € j- [± I - sin € _ -f + cos c — /- as \dyj as dx ds dy <]1 dx 2 3' dxdy and at 0, **-&+*& ds dn dx dy ds dy or d 2 f _ d df 1 df dxdy ds dn p ds Similarly from the second of (i), ds 2 dy 2 p dx or c ^ f = c El+ lc ^ dy' 2 ds 2 p dn (iii) (iv) Thus the values at of -j-. -/, ^-A-, t4 are known when /'and /- are given along dx dy' dx ay dy 1 J dn ° & the boundary. ~d~ 2 + ct 2 or v!/* Dem g an invariant for all systems of rectangular axes, we may also conveniently take <iy = 2f _i_tf_c£f i/.r- p dn ds- (v) 1. Equations of equilibrium. Form of solution for a plate free from bodily force. The equations of equilibrium of a homogeneous isotropic elastic solid are of the form xx d xi/ 'lx dy d i d xij lx d 7:. dx dz + X = d yy ^'dyz + y = Q dx dy dz d yz d & v + - — + - + Z dy dz (1) xx, yy, zz, where X, Y, Z are the components of the bodily force per unit volume, and xy, xz, yz are the components of stress, these being given in terms of the displacements u, v, iv by the equations xx — AA + 2/x yy = AA + 2/x \ a « dw AA + 2/x— where ~ (dw dv yz = *\Ty + & ^ fdu dw\ ,z = \dz + dx) — (dv du\ dz~ ' xy = ^{dx + dy) du dx b: \ (2) A = an du dw fx dy dz THE EQUILIBRIUM OE AN ISOTROPIC ELASTIC PLATE 141 In terms of the displacements, the equations of equilibrium are therefore MV 2 M + (A + M )-, A + X = dx HV 2 v + (* + A*)^r + Y = . dy yu.V-W + (A + /x) + Z dz (3) (i) (») When the bodily force is null, or X = Y = Z = 0, the following forms are easily shown to satisfy equations (3), "dy\ dd\ dz dO dy 06 dz \ + % l xd<t> + il? d i <j> \ A + /A dx dz dx A + 3/A c?c/> 2 rf 2 <£ k + jx dy dz dy (iii) m = w = A + 3/A <7<£ 9 r7 2 (/> A + p dz where ^, 0, (p are potential functions, so that vV = °> v a # = 0, v 2 <£ = . These solutions have been used by Boussinesq in his treatment of the problem of a solid bounded by a single plane z = 0. They are equally effective when the boundary consists of two parallel z-planes. Thus, as will explicitly appear in the sequel, and as might be proved at once, any solution of (3), with X = Y = Z = 0, in the space between the planes z= ± h, can be expressed in the form n dil/ dO dd, , d~4> \ u = 2 r + — + a-f + 2z— -? dy dx dx dz dx , v = -2 d A + d l +a ^ + 2z^ dx dy dy dz dy w = dO dz dd> n d 2 d> i-^- + 2z— -£ dz dz 2 J (4) Here, and throughout the paper, the symbol a is used to denote the fraction (X + 3m)/ (a + m). With these values of u, v, iv the stresses across a z-plane, viz., zx, zy, 7z, are given by zx 2/JL 2/, <m . . - ~-^r + 2z- d 2 4> d 3 <f> ) d 2 ty dy dz "^ dx dz ' dx dz dV ^ d*6 ^ d 2 <f> dx dz' 2 dx dz dy dz ' dy dz d 2 6 d 2 <j> ~dz 2 + 2z dy dz 2 + 2z^±- dz 2 dz s (5) 142 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 2. Force applied at a single point. Returning to the solutions (i), (ii), (iii), we note that (i) and (ii) contribute nothing to the dilatation A, and (ii), (iii) nothing to the z-rotation a> s = -^(-^ - J-) ■ These properties can be used to resolve any given displacement into its ^, 0, (p components, the bodily force being null. An example of fundamental importance is the displacement in an infinite solid due to a single force applied at a given point. Thus for a unit force applied at the origin in the direction of the axis of z we have u \ vrz V =..y z ,.s w 2 2 a = -S + - ► each multiplied by — 1 \+jl _ where r 2 = x 2 + y 2 + have 8-n-fi A. + 2/a 47r / a(a + I ) or say, for a Z force of 4-717* (a+ 1) units applied at (x\ y' , z') we dr~ l U = (z' - Z) v ={z'-z) dx dy dr- 1 v= (z'-z)%- +ar-i dz (6) r x being written for l/r, where r is the distance from (x, y, z) to (x', y', z'). These give But in (4) A= (a-l)_ ; <o, = 0. dz A = 2(l-a)ft;"3 = ft dz- dz- Hence we take ^ = 0, and choose (p so that d<f> dz" 2 ? Now the functions log [r+z — z') and - log (r -z-z') are both potentials having r' 1 for z-derivative ; the former is without singular point in the region z > z', the latter in the region z < 2'. We may without confusion use a single symbol to denote either function indifferently, and define dz~ We may therefore take r = log (r + z- z) when z > z \ = -log (r -z + z) when z<z \ (7) 1_ d- l r~ l J ~dz~' Comparison of the displacement w in (4) and (6) gives now d$ ,dr = z — + ~ r- 1 . dz dz 2 THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 143 For a Z force of 4-^(01+ 1) units at (x', y f , z') we have therefore t/r=0 , _! a d V 2 cfcr 1 1 d" 1 /-" 1 ^~ 2" "dJF 1 " (8) It is easy to verify that these values of >!/, 0, <£ substituted in (4) do actually reproduce equations (6). Similarly for an X force of 47r / u(a+ l) units at (x\ y', z') we find _a+l c2 d 2 r l *~ ~2~~ Ty dz~ 2 Z T- 1 rf dr'-r- 1 4> = ^ 2 dz dz~ 2 (9) Here — 2 - denotes a potential function having -j 7=r for z- derivative, and is defined by the equations — — = (z- z) log (r 4- z - z) - r when z>z [ /, ~v = — "(2 - z') log (r - z 4- z') - r when z<z' J It may be observed that the necessity for dealing separately with the two regions z>z' and z<z' in these cases is not inconsistent with the theorem of (4), which refers only to a displacement free from singularity in the space considered. 3. Solution of the problem of normal traction. Coming now to the problems relating to a solid bounded by the two parallel planes z — h and % = - h, we begin with the simplest of these, and seek a solution of the equations of equilibrium giving X — Y = Z = throughout the hody ; the normal stress w=f(*,y) on z = h, = on z = - h ; the tangential stresses zx, zy = on both faces z = ± h. The arbitrary function f(x, y) , which we shall suppose to vanish at all points without a given finite area A, is expressed in a form amenable to analytical treatment in the familiar theorem Limit f ( */(*', y')dx'dy _« />„ a ni x the integral being taken over the area A. (If we imagine the plane z = to be covered with attracting matter of surface density f{ x > 2/)> tnen the theorem expresses the well-known relation between the density at 144 MR JOHN DOUG ALL ON AN ANALYTICAL THEORY OF (.'•, //, 0) and the limiting value of the normal attraction at (x, ?/, e) as e approaches zero. ) As a preliminary to the general problem, we take therefore the special case in which « on z = h is equal to e/{(x — x'f + (y — ;/) 2 + e 2 }- , or, in the form of a definite integral, I c-^kJi/kR)^, R 2 = {x-xf + (//- 2/')2. where Making a further reduction, we begin by taking, in place of this integral, simply the function kJ /cB,. The function ^ is not required, and <p, are of the forms <£ = (C\ sink kz + C. 2 cosli kz) J kR i 6 = (C 3 sink kz + C 4 cosh kz).T kR J In accordance with (5) these satisfy the conditions (16 , dd> , n '7 2 (& A . , — + — '- + 2z — i =0 on z = + li dz dz dz* dM_dty + 2;/ P<jJ = Q on z= _ h dz 2 dz' 2 dz 3 = kJ kR/2/x on z — h (12) Hence we easily find (13) - . cosh kIi. t t> ■ i 4urf> = - - . J,,kK sin i k? rr *(sinh 2kA - 2k/i) ° sink /c7i T -r, i — ; J „kK cosh kz K(smh 2k1i + 2/c/i) ° , /, cosli k/i + 2k/; sinh «// T -r> ■ i ^fl^ -. . , , , —-^r J„kR sinh kz r K(smli2K/t-2K/i) ° sinh kIi + 2k//. cosh k/i t „ , + J,,kK cosh kz K (sinh 2k/i + 2kIi) ° If these expressions, multiplied by e"* 5 , could be integrated with respect to k from to oo , we should have at once a solution of the preliminary problem. But this integration is not possible, owing to the nature of the functions of k near the lower limit k = 0. In fact, if the values of 4-.fi.ip, 4fi.6 in (13) be expanded in ascending powers off, the expansions will contain terms in 1/k 3 and l/«r, so that near «• = 4fx<p = H/k 3 + K/k + terms of positive degree 4/x0 = L/k 3 + M/k + These terms of negative degree are potentials contributing nothing to the stresses on «=± h, as we see from (12), since kJ «:E contains no terms of negative degree. They might therefore be subtracted from the expressions (13) without affecting the satisfac- tion of the conditions in (12). This simple subtraction would, however, introduce terms not integrable right up to the upper limit, at least after e is put equal to zero, as eventually it will be. The difficulty is met by subtracting from 4fi<p, not R/k 3 + K/k, but ri> :i -r-Ke- K 7/c; and from 4^9, not L/k 3 + M/k, but L/^ + Me-"*/*. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 145 (There are, of course, any number of equally suitable modifications ; instead of e' Kh we might take e~ KC or l/(l+/c 2 ), for instance.) A solution of the preliminary problem of normal traction equal to <?/(R 2 + e 2 ) 3 on the face z = h is thus obtained in the form 4/*<£ = - / e- ™\ . "^ K l J kR sinh kz Jo ( K(sinh 2k/i — 2k1i) sinh ah T -n i H Ke _,cft | , + / • i « 7 =r-rT J oK K cosh kz - — > a.K k (sinh 2/Ji + 2/c/i) k 3 k J f cosh /c7i + 2/Ji sinh k7i-, \ k (sinh 2kJi — 2/Jt) , sinh kIi + IkIi cosh k7i t -t, i L Me~ Kh ) -. ,-,,, + — /.-,«,■, — s^rr J o K -K cosh kz -— > cIk • • (14) k (sinh 2»c/t + 2k7i) k 3 « J v ; The solution of the original general problem is found by multiplying by f(x', y') 1 2ir, integrating with respect to x', y' over the area A, and finally taking the limit for e = 0. But a glance at the forms near k=co of the functions in (14) shows that the triple integrals are absolutely convergent, it being supposed that — h<Cz<.h. Hence we may integrate with respect to x', y' first, and by a well- known theorem the limits for e = may then be found by simply putting e = in the integrands, provided the resulting integrals are convergent, as they manifestly are. This gives the value of (p, for example, in the form /„ dK j fA z '> y'M*Wdy', but, always provided -h<.z<ih, we may change this if we please into fff(x',y')dx'dy'rj(K)dK. Finally, we may with great advantage confine our study in the first place to what is usually spoken of as a unit element of normal traction at (x', y', h). The area A enclosing this point is diminished, and the intensity of traction increased without limit, so that j ff(x', y')dx'dy' remains equal to unity. The resulting solution is simply that of (14), but with e put equal to zero within the integral signs. As we have just seen, the solution for the general case can at any time be found from this elementary solution (15) by multiplying by f(x', y')/2-7r and integrating over the area A. 4. Flexural and extensional components of the strain. Disadvantages of the solution in definite integrals. In the elementary solution each of the potentials (p, 6 may with advantage be decomposed into an odd and an even part in z. Thus, for an element of normal TRANS. ROY. SOC. EDIN, VOL. XLI. PART I. (NO. 8). 24 (16) (17) 146 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF traction of 8^^ units at (x', y' , h), a solution is given by <j> = <j) + </>,, ; 8 = 6„ + 8 e where , /"" j coah kJi _ _ . . 3z e--" fe /2z 3 -3R 2 z y o I k (sinli 2*/;. - 2k/&) ° 4x i 7t 3 » r* i cosh kJi + 2 kJi sinli kJi T t> • i 3z e~' 6 « = i — / • uo i — nr j ° kR sinh * z ~ T-373 ~ — .' i) I k (smh 2k/i - 2k/;) 4^/^ k A = I \ - -_ -Jl"—— J kR cosh kz + — — [ cLk ^ Jo 1 K(sinh2K/; + 2K/;) ° 4k j /, /"" I sinh k/; + 2k/i cosh k/( T -r, , Se-" 71 ) 7 6 = 1 { J ( ,KKcoshKZ- > fiK )» I K(sinh 2k/i + 2k/i) 4k I The conditions satisfied at the faces by the partial solutions (16) and (17) are easily made out. For when <f>, 9 are both odd functions of z, then «e, zy are even and zz odd ; but when <j>, 6 are even functions of z, then z~x, 7y are odd, 7z even, as is obvious from (5). Hence (16) gives equal values of opposite sign for 7 at corre- sponding points on z= ± h ; (17) gives equal values of the same sign. It follows that (16) is the solution for elements of normal force of 4717* units at each of the points (x\ y', li), (x', ?/, — h), the force being in the positive direction of Oz in each case, and therefore a traction on z = h, but a pressure on z= — h; in (17) the only difference is that the force is a traction on both planes. Hence, also, (16) subtracted from (17) will give the solution for traction on z= —h alone. Each of the integrals in (16), (17) defines a potential function without singularity at a finite distance in the space between the planes z = ~b %, and all the successive deriva- tives with respect to x, y, z of any of these functions may be calculated by differentia- tion within the sign of integration, provided we are dealing with a point actually within the solid, so that — h<z<Ch. The solutions defined by these integrals are therefore formally satisfactory. It is, however, a serious objection to them that they do not lend themselves readily to inter- pretation, and it is not easy to make out from them any of the simple laws which the ordinary approximate theory leads us to anticipate. In particular, the solutions in their present form throw no light on the question of the behaviour of the functions and their derivatives at points the distance of which from the sources of strain is great in comparison with the thickness of the plate, a question of great importance for the application to the thin-plate theory. The analytical transformations to which we now proceed reduce the solutions to a form entirely free from these objections. Each of the integrals is shown to be composed of two parts of very different character. The first part represents a function the value of which diminishes with great rapidity as the distance from the source increases, while the remaining part is a function of very simple form. Each solution is thus resolved into a permanent or persistent element and a local, transitory, or decaying element, the latter being insignificant beyond the immediate vicinity of the source. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 14f 5. Transformation of the definite integrals into series by means of Cauchys Theorem. The integral (p of (16) can be written as the sum of three integrals, namely — [" p j cosh kIi sinh kz 3z 1/z 3 9 z\ I , *' = J o JoK 1 K(sinh2^-2KA) IkW K \m TQhJf )£(J>B-e-*)- (18)^ _f_ _9 z\r,T t, . „^^ 8/i 3 40 fcy It should be observed that the first and third of these integrals cease to converge when K = 0. Hence the transformation does not apply to points on the line x = x', y = y', the normal to the plate through the sources. Consider now the first integral in (18). The function of k multiplying J kR within the integral sign is an odd function of k vanishing for k = 0. Hence, as in (i), the integral is equivalent to the complex integral f ' r t? / cos ^ ^' s ^ n ^ KZ 3z 1/ z 3 9 z\ \ , J S ° I k (sinh 2Kh--2Kh) + 4^/T 3 + AM 3 40 h) J ' the path of integration running from west to east along the whole of the real axis, and just avoiding the origin, which is a singular point of G kR, on the north or upper side. On this path take points E, W at distances mr/2h to the right and left of the origin, and on E W as side describe a square E W A B in the upper part of the plane. The integral over each of the sides WA, AB, BE is easily proved to have zero for limit when n tends to infinity through positive integral values. Hence, by Cauchy's fundamental theorem, the integral over the path W E is equal to the sum of the residues of the integrand at its poles in the upper half of the k plane, multiplied by 2iri, that is, to the series Zn -r>/ \ cosh kIi sinh kz G o* R ( - ) -T7 — uin T\ ' k *A(cosh 2k/i - 1) the summation extending over the zeroes of the function sinh 2*h — 2*h in the upper half of the k plane, in the order of their moduli. If £ n is a zero of the function sinh £— £, the corresponding zero of sinh 2K.h — 2«h is K n = % n /2h, and G /c, l R = G |^- = /s y^— e i e 2h approximately, and we see that this part of (p , with its successive derivatives, is practically insensible when R is a very moderate multiple of 2h. (Cf. § 7, infra.) 148 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF As to the other two integrals which occur in (18), we have proved in (h) that These functions will occur so often that it will be convenient to reserve an invariable symbol for the former of them, say X (R), or simply x , = -R 2 log — - ^R 2 and then V 2 X = lo 8; R 2h (19) The persistent part of <p is therefore 3z Z 3 . 9 z 4A 3 * + W + 40 h) V ~ X ' which is the sum of two potential functions, 6. Types of the particular solutions composing the general solution. A glance at the relation between the results just obtained and the form of <£ in (16) enables us to write down at once the corresponding transformations of 6 , (p e , 6 e . Collecting the results, we find , y n p/ > cosh kIi sinh kz 3 / 1, , \ 9 «, ^ = zL G ,R( - ) Kh(cQah2Kh _ i ) ~ m (*X ~ 6 *V X ) + m ^x 6 n = ZG KR^ cosh Kh ' + 2 f sinh " 7 ') sinh KZ + ±(z x - l 2 3 v2 _ 2/fev 2 ) _ 9 2 ■ ° K7i(cosh 2k1i - 1) 4/A x 6 V X V X J 40/, V X where k is a zero of sinh 2kIi — 2/ch, with positive imaginary part. (20) *.-2g *B(-)-? sinh kIi cosh kz + lV 2 X kJi (cosh 2 K h + I) 4 n _ p T?(sinh kJi + 2kJi cosh kIi) cosh kz _ 3 ., ~ « ° K k)i (cosh. 2 K h + I) ~i Vx (21) where k is a zero of sinh 2kJi + 2k1i, with positive imaginary part. The solution must give w, zy, 7z all equal to zero at the two plane faces of the plate, except when R = 0, and we are thus prepared to find that the strain defined by the terms corresponding to any one root k gives zero stress across z = =fc h. Thus in (20) 2 contains a series of particular solutions of the type (i) 4>= - cosh k?i sinh kz ~F(x, y) i where (y 2 + k 2 )F = \ 6 = (cosh kIi + 2kJi sinh k]i) sinh kz F(x, y) ) sinh 2kU - 2k1i = J (22) THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 'Calculating the stresses by means of (5), we find - — + — ® 4- 2z — f = (2k 2 7i sinh kIi cosh kz — 2k 2 z cosh nfi sinh kz)F dz dz dz 2 — - — * + 2z — f = 2k' 2 (cosh k/i + kIi sinh «7i) sinh kz-~E - 2k 3 z cosh k/i cosh kz-~F dz 2 dz 2 dz 3 v ' both of which vanish when z= ± h. We have further in (20) a solution of the type 149 (ii) *=- 2 F + ^VF From this, by (4), (5) = ^f4*VF-2^v 2 f K : ' ■" ,h '""' ° W = d ( dx w = — w = (a+l)(F-i2VF) + 2(^ 2 -A 2 )v 2 F 9=4 / ,( 2 2 -/^V 2 F (23) The solution for unit normal traction on z = A contains a strain of this type, with F = X (R) • 3/32ttM 3 . Lastly, in (20) there is a solution of the type (iii) fj> = — zF ) Fa function of x, y zY) = - (a + 1 ),v with v 2 F = , d¥\ dx w = (a+l)F ; Obviously this is merely a degenerate case of (ii). Again in (21) we have a series of solutions of the type (24) (iv) 4> = - sinh K.h cosh kz F(x, y) ) where (v 2 + k 2 )F = ) 6 = (sinh Kh + 2k/i cosh */<) cosh kz F(:c, y) / sinh 2/</* + 2k/< = f (25) In this, as in (i), z * = zy = zz = on 2 = ± h. (v) The persistent part of (21) is of the type giving 4> = F(z, ?/) ; = - 3F(z, v/), where v 2 F = ) ,eZF , .,^F ~ ~ " = {a ~ 3) ^' v = (a " 3) ^< ; ^ = 9 = s = ) (26) 150 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 7. Position of the zeroes of the functions sinh£=b£ The nature of the infinite series occurring in the above solution will be made more intelligible by a short discussion of the position of the roots of the functions sinh2x-/i±2'c/i. These are obviously found from the corresponding roots of sinh £±£. by dividing by 2h. (i) sinh£-£=0. £ = is a triple root, and the remaining roots are all complex, falling into sets of 4 of the form ±p±iq, where p, q are real. If, then, £=£ + *>?, £ and n being real, we need only consider the case of £ and >; both positive. We have then sinh $ cos 17 = £ and cosh £ sin 77 = 77 . Cos n and sin 1 are therefore both positive, and n must lie between 2mr and 2nir 4- 71-/2- It is easy to prove that there is no root between and 71-/2. For £ > tanh £, or £/ sinh £> 1/ cosh £, so that cos 77 > sin 17/17 or 17 > tan 17 or 17 > it/2 . For every positive integral value of n, however, beginning with n—1, there is one root, and one root only, with n between 2n-rr and 2mtt + x/2. This will be readily seen on roughly tracing the graphs sinh £ cos n = £ and cosh £ sin 17 = 17, or it may be proved by an elementary application of the Theory of Functions. Thus, if we make the variable £ describe the contour of the rectangle formed by the four lines 77 = 2w7r, f=N, i7 = 2ra7r + 7r/2, £=0, where N is a large positive number, it will be found that the function v = sinh£ — £' describes once a contour enclosing the point v = in the v plane. There is therefore- just one point within the rectangle at which v becomes zero. For the large roots cos n must be small, or 77 = 2ra?r + tt/2 - e, where e is small. Hence cosh ^ = r) — 2mr + tt/2 , or £ = log (4ra+ In-) approximately. Then e = £/ sinh i=2 log (4?i + 1 ir)/(4re + 1 tt). By successive approximation we may now find the roots as nearly as we wish, but exact values are not at all necessary, the first approximation being quite sufficient for our purpose, £, = lQg(4n + lir) + C2n + i>rf .... (27) (ii) sinh£+£=0. In this case £ = is a simple root, and the rest of the roots are complex. If %=$i + iy, we have sinh £ cos 17 + £ = and cosh £ sin 17 + 17 = . Cos»7, sin *j are both negative when £ and *? are positive ; hence >/ lies between (2n— 1)tt THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 151 and (2n- 1)^ + ^/2, and it may be shown, as in the previous case, that there is actually •one root with *i between these limits for all positive integral values of n. Also 7] = (2n - 1 )tt + 7J-/2 - e ; cosh £ = 2?nr - tt/2 e = 2 log (4re - 1 t)/(4» - 1 7r) . To a first approximation £„ = log(4ra-lTr) + (2ra-£>ri .... (28) In addition to the roots of (27), (28) we have of course a corresponding series in the second quadrant, the images of these in the axis of imaginaries. 8. Approximate forms of the n th terms of the infinite series, when n is large. It may be useful to give in terms of n approximate forms for the general terms of .(20), (21) corresponding to the n th roots in the first quadrant, (i) (p and . kIi = \ log 4?«7T -i- (n + \)iri sinh kJi = \e Kh = ■£(4?^7^)ie( n +}> , ™ cosh kIi. 1 4 -e-( M +lM( -i) «7i(cosh 2<c/i - 1) sinh kU ■ 2 sinh 2 K/t (4?i; z z sinhxz = l(e Kh )h -\(e Kh )~h z z ( — TTlZ -, TXZ Hence in <p the general term m(z-h) *t* m(z+h) 4 i *—!l TTiiz-h) ZJ4- m(,z+h) ) = -J- G kR \ (4n*r) 2A e<»+i>— £— - (4«7r)- 2A e-(»+i)— IT" S ; in O , the same as this, with the factor i/2mr omitted. In both G„»cR = f-—e- (»+Dx e 2A (ii) (f) e and e . kIi = £ log 4ra-7r + (w - \)iri cosh K h = i(4?jTr)ie( ,l -i)' r,: In <p e the general term h^-G " r 1 (4«7r) a*e(»-J) \ +(4ra7r)- 2A e-("-l)—ft— V In O the same, with the factor i/2wr omitted. , 2riR 9. The solution for arbitrary normal traction. Questions for discussion. The solution of the general problem of given normal traction requires the multipli- cation of the functions in (20), (21) by f(x', y') and integration with respect to x', y' over a finite area A. There is no difficulty in showing that these integrations can be performed term by term, and that the resulting series converge absolutely. 152 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF When the solution of the general problem has thus been obtained in terms of series of surface integrals, several questions present themselves for treatment, among which may be specially mentioned (i) For how many times in succession may these series be differentiated term by term with respect to the coordinates x, y, z% (ii) When the thickness of the plate is infinitesimal, but /(a?, y) does not vary as h tends to zero, what are the orders of the various parts of the solution, and of the related physical quantities ? (iii) How are the answers to these two questions affected by discontinuity in the applied traction, or its x, y derivatives 1 A perfectly general discussion of these questions would be tedious and difficult, and it will probably be more useful to consider the points suggested in the light of a special case, in which the integrations required can be performed, and the outstanding features of the solution can be grasped with comparative ease. 10. Detailed solution of a special case. Term by term differentiations. The solution we propose to work out is to satisfy the following boundary con- ditions : — zz = 4:TTfiJ m (Pp) cos mm > on z = h = - 47ryw.J„,(/?p) cos mu) , on z = - h — oii2=+/[, when p > a zx = zy = , on z ± h . > when p < a p, w, z are the cylindrical coordinates of the point (x, y, z), so that x — pcoso), y = p sin w. /3 is any constant, and m is an integer. The solution, obtained from (20) by integration, is 4> = <fy } + 4> 2 ; = 6 l + 6. 2 , where *>= i( 2F -l zVF - 2 "^ !F )-jo^ 2F • ■ <M> with F = I /x(R)J„<(/V) cos mw'p'dp'ddi , V / \ C0Sn kJ ' SiIin KZ f fn -D T O ' ' 1 ' 7 ' 9-2 = Z-l \ ~ ;-r; ^ — ;— ; rr I \jc,k\sA m ap cos m<a p dp day a 'V (cosh k/i+ 2kJi sinli kIl) sinh kz , • , ,, #2 = ^-i , , --.-« -, /, • (same mtesral) (30> the integrals being taken over the circle of radius a. Consider in the first place the part of the solution defined by (p 2 , 6. 2 . The value of the surface integral in (30) takes different forms when p > and <«, THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 153 As proved in (c) 9 9 when p<a, the integral = , i) u7r J m Bp cos mo> + „„ "* J m Kp cos mco(K«G m V<J„,/3tt - Q m KafiaJ „'(3a) /J- - K l ft' - K" p>a, ,, = & m «p cos vio)( kuJ m r Ka J nl /3a - J m KaBaJ m '/3a) . P 2 - K Z Now when up, kci are both large, J mK aG m K P = — L_ e >;*(P-«), 2 k ,Jap which, with its derivatives, is very small when (p — a)j'2h is even moderately large. Thus, in the space without the cylinder p = a, the part of the strain given by <p. 2 , 2 is, when h is small, insensible except in the immediate neighbourhood of that cylinder. The same remark applies to the strain within the cylinder, so far as it is given by the parts of (p 2 , &> arising from the second term in the value of the surface integral. We naturally inquire, how do these rapidly decaying parts of the solution behave, and what is the order of magnitude of the corresponding displacements and stresses, at points actually on the surface p = a1 Now, taking for example the value of (p 2 in the external region, namely, <ft 9 = 2-i ( - ) f°. S 1 " \ S1 f ^ f x TZ—- y &™ K P cos m<a ( KaJ m ' K aJ m /3a - J m Ka[3aJ m ' fta ) • • (30'} k kIi (cos1i2k/i - 1) p~ — K" \ / we see from § 8 (i) that when p = a, the general term has the approximate form A ( z Jlh. m(z-h) z+h irtiz+h) ) A being independent of n. Moreover, each differentiation of (p 2 with respect to p or z will remove a factor l/n from the general term. Hence three such differentiations, but no more, are permissible, if— /^<z</i. But from (4) it is clear that none of the displacements requires more than two, and none of the stresses more than three of these differentiations for their calculation. As for 6 2 , the general term is of one order higher in n than the corresponding term in <p 2 , but in compensation for this only two differentiations are required to find the stresses. Hence, so far as the decaying part of the solution is concerned, displacements and stresses at p = a may be found by means of term by term differentiation, and subsequent substitution of a for p. Again, considering the order of these various quantities in h, regarded as infinitesi- mal, and remembering that kJi and kz are of order zero in h, we see that the expression for <p 2 in (30') and the corresponding expression for 9 2 are of order h 2 when p = a t and each differentiation with respect to p or z diminishes the order by one. Hence the displacements at p = a are of order h and the stresses of order zero, so far as they arise from the decaying part of the solution (p 2 , 6 2 . TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 25 154 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 11. The same special problem. Summation of two infinite series. An important part of the strain given by <p 2 , 6> 2 remains to be considered, namely, that which arises from the term 27r/(/3 2 — k 2 ). J m ftp cos mw in the value of the surface integral for the case when the point (p, w) is within the cylinder p = a. Denoting these parts of <p. 2 , 0. 2 by <p. 4 , 3 , we have <£ 3 = 2TrJ,„f3p cos mm £j( — ) cosh kIi sinh kz (f3 2 -K 2 ) K h (cosh 2k/4-1) _ , ,, "V (cosh k/i + 2k/< sinh */t) sinh «,? k (/}- - k-)k/i(cos1i 2/</i - 1) (31) We note in the first place that <p 3 admits of three, and 3 of two term by term differentiations with respect to z when — // < z < h, while x, y differentiations can be performed without restriction. Combining this result with those already obtained, we see that in the complete solution in terms of surface integrals, all the differentiations necessary to give the displacements and strains or stresses at any point in the body of the plate can be performed on the series term by term. When h is small, <£ 3 and 3 are of order h 2 , and a z- differentiation lowers the order by one. This can be seen from the series, or otherwise, for, as we shall now show, the value of the series can be found in finite terms. Consider the function of k, cosh kIl sinh kz {j? - /c>(sinh -2kIi - 2 K h)' This function, multiplied by k, vanishes at infinity at all points of the path E W A B described in § 5 ; hence the sum of its residues vanishes. The function being odd in «:, the residues at the poles k = db k 1 are equal. Thus 2 (series of residues at zeroes of sinh 2kIi — 2kIi in upper part of plane) + 2 (residue at k = fi) + (residue at k = 0) = . The residue at k = ft is / _ , cosh fih sinh fSz '2/3-(sinh 2/3h - 2f3h) ' Also if cosli k/i sinh kz A B , „ = -=- + (-•••• , near k = k (sinh 2kJi — 2kU) k 3 then the residue at k = is Hence 2, i _ \ cos ' 1 "h s ' nn KZ _ cos h Ph sui h P z , A B « ' (/J 2 - K 2 ) K h (cosh 2<h - 1) ~ y8 2 (sinh 2(3h - 2/3h) /^ + /F' It may be noted that A/ ft 4 + B/ft' 2 are simply the terms of negative degree in the expansion of cosh j3/t sinh f3z P 2 {smh2ph-2/3h) in ascending powers of ft. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 155 Hence, putting in the values of A and B from (16), (18), f _ cosh fflt sinh ftz 3z 1 / z 3 9 z \ \ <£ 3 = 2wJ fB #> cos mm <j ^(^2^-2^) 4/3% 3 ^ \8P 40 ¥/ J and similarly „ T „ ( (cosh Bh + 2Bh sinh Bh) sinh Bz 3z 1 / z 3 , 69 z \ \ e 3 = 2,J m B P oo S m,^ BHs^2Bh-2Bh) -m^-Ash^TO-hJl (32) 12. The same problem. Final form of the solution. We come lastly to (p 1 , X of § 10. The function F requires separate formulae for its expression in the three cases m = 0, m = 1, ra > 1, but in all cases F = (2Tr/B i )J m Bp cos mm + F 1 , when p< a ) F = F 2 , when p>a ) where F x and F 2 satisfy the equations v 4 5\ = , y 4 F 2 = 0. The values of F a , F 2 for the various cases are given in (/). When p<a , the term (27r/l3 i )J m 6p cos mw of F , taken by itself in (^ and l9 would give _3_ ^_JVz 3 1 Al ^ 4/i 3 /3 4 y8 2 \8)i 3 + 40 fc. 1 4/t 3 /3 4 + /3 2 W 3 + 40 ft,/ J •2ir3 m Bp cos raw . These are precisely the terms of negative degree (both in /3 and in h) with signy changed, in the expressions for cp 3 and 6 3 given at the end of § 11. If, then, we take this part of (p t , X along with cp 3 , 3 we have the complete solution in the form + 2 7 r2(-) cf> = - 2tt J m Bp cos mm cosh K.h sinh kz cosh Bh sinh /3z £ 2 (sinh 2/3ft - 2/3ft) 08 ^ ^— — — — — : — - J m «p cos mm(KaG m 'i<aJ m Ba - G mK aBaJ m ' Ba) 1 - K z )Kh(cosh 2k/i-1) 6 = 2irJ m Bp cos mw (cosh /3ft + 2£ft sinh /3ft) sinh /Sz and /3 2 (sinh 2/?ft - 2/?ft) "V (cosh k1i + 2kIi sinh «ft) sinh kz t , ~ , T Q n T <a \ ' 2 "-f ( /j'-,.'H(cosh2,;'-i) J -"' cos '"™ ( "' G - "< ,J " /3 '' " °- M /3 " J " W + U z *> -r VF - - 2,A ' !F - 4^ when p<a . (33> * = 2ir2(-) cosh «ft sinh kz (/3 2 - /c 2 ) K /i(cosh 2,<ft - 1) _8_ ( 4ft 3 G m Kp cos mm(i<aJjKaJ m Ba — J m KaBa 3 m 'Ba) *.->*».) + j^. /i o "V (cosh k.1i + 2k1i sinh k/j.) sinh kz~, / t ' t o t „ o t 'o~\ ^ = 2tt^ v — 77 - — — — — —L — — — G mK p cos mm{KaJ m KaJ m Ba - J„,Ka BaJ m Ba) K (B - K 2 )Kh(cosh 2k.1i- 1) + 4-JI-K «--.-j«^ p .-"^0-jjK^ p » when p > a . (34) 15C Mil JOHN 1KHJGALL ON AN ANALYTICAL THEORY OF In (33) and (34) each line represents a potential function ; in (33) the first lines define a particular solution giving the proper values of the tractions at the surface, as may be seen from (13) ; the partial solutions given by the k series give zero surface tractions, and represent a strain insensible except in the vicinity of p = a ; and the solutions defined by the last lines, being of the form (23), give zero surface tractions. From these remarks it follows immediately that the solution (33), (34) satisfies all the conditions of the problem in the two regions p < a , p > a , taken separately. To verify the solution completely, it would be necessary to show in addition that certain conditions are fulfilled at the surface p = a, namely, (i) that the displacements and strains are continuous at this surface, and (ii) that the integral value of the stresses ex, zy ,%z over any small area lying partly within and partly without the cylinder p = a, on either of the plane faces of the plate, tends to zero when the area is indefinitely diminished. The condition (i) ensures the ' synexis ' of the solution across the surface p = a, and can be proved by showing, as may easily be done by means of summations similar to those of § 11, that (p , 6 , -=r and - T - are continuous at that surface. For by the Theory dp dp of the Potential this carries with it the continuity of all the derivatives of <p and , and therefore of the displacements and stresses, as well as of all their derivatives, under the proviso, of course, that — h < z < h. The condition (ii), or some equivalent, is required in order to exclude the possibility of stresses with finite resultant passing into the solid through the lines z = db h , p — a ; or, in other words, in order to ensure that the solution is not partly due to linear elements of traction at these lines. 13. Order of the various parts of the solution, ivhen h is small. The final form of the solution, as exhibited in (33), (34) was obtained by combining parts of <p x , #-, with <p. 2 , 6 2 , and until this was done, it was not immediately evident that (p and were potentials. Thus the part of the solution arising from the imaginary values of k, or from any one of them, is not, within the region of applied traction, a potential by itself, and the same is true of the (p-^ , 1 part, which may be considered as coming from the zero values of k. This has sometimes to be taken into account in calculating the stresses ; the formula for zz, for example, in (5) requires additional terms if, while u, v, w are still given by (4), (p and 6 are not potentials. On the other hand, the separation of the solution into the two parts (29), (30) has this very marked advantage that, when h is very small, the first part gives the terms of the two lowest orders in h of <p , , namely those of orders h' 2 and h°, while the second part, as we have already seen, contains no terms of lower order than h 2 . When, how- ever, we come to calculate displacements and stresses, the separation is less simple, mainly in consequence of the fact that x , y differentiations do not change the order of <p } , 1 , but diminish the order of <p 2 , Q., by one for each differentiation. THE EQUILIBRIUM OK AN ISOTROPIC ELASTIC PLATE. 157 The following table, which may be deduced immediately from the results of SS 10, 11, shows the order in h of displacements and stresses arising from 2 , $ x and (p., , 2 respectively. U , V IV xx , xy , yy ZX, ZIJ «£ 2 » #2 *!.*! p2« p = a -2,0 2 1 -3, -1 1 1 -2, -1 1 It thus appears that the first part of the solution gives all the displacements to a second approximation, and all the stresses but zz to a first approximation. With regard to T z , it should be observed that the solutions depending on F x , F 2 contribute nothing to it, so that, within p — a, its value comes altogether from the particular solution, and without p = a, its value is zero beyond the immediate vicinity of that surface. 14. Methods and results of the special case extended to the general problem of arbitrary normal traction. One feature of the solution expressed by equations (33), (34) we have already found useful, especially in the important case when h is small, in such a way that fih and h/a are small fractions. We refer to the explicit separation in the solution of a purely local element, entirely negligible except within a certain strip of breadth com- parable with the thickness of the plate, from an element of a persistent or permanent character, with an area of influence not affected by the indefinite diminution of h. Another advantage of the form of solution in (33), (34) is that the particular solution for the space within which the traction is applied is found in such a form that it can be readily expanded in powers of h, so as to give the terms of positive order in the infinitesimal h, as well as those of negative order which were already separated in (29). Thus in the particular solution, or first line of (p in (33), the factor cosh [ilu sinli f3z /3 2 (sinh 2/3/;. - 2 /3k) can be expanded in ascending powers of (3, the series converging if j 28h | < | ^ | , where ^ is the complex root of sinh £— £ = with smallest modulus. Since z is of the same order as h, and we are supposing /3 independent of h, it is clear that the terms of the series will be of ascending order in h. We shall now show how the solution for the general case when the given normal traction is a function of x , y of unspecified form may be transformed, under certain restrictions, so as to yield the advantages to which we have been referring as pertaining to the solution (33), (34). The problem we suppose to be the same as that stated at the beginning of §10, 158 MR JOHN DOTJGALL ON AN ANALYTICAL THEORY OF but with f(x , y) instead of J\„fip cos ma , and with any continuous area A instead of the circle within p = a. The solution will then be of the form defined in (29), (30), but the integral of (29) will now be f = ffx(W(*, y'Wdy' and the integral of (30) I = jJG (KR)f(x',y')dx'dy: Since ^ 2 x (R) = log(R/2h) we have V*F = V 2 (V 2 F) = 2*f(x,y)\ also (v- 2 + « 2 )I= -W(x,y) ] W If f(x , y) and its derivatives of the first two orders are finite within A, we may transform I by Green's Theorem. Thus, excluding from the area A an infinitesimal d 2 d 2 circle about (x , y) as centre, and writing v 2 for ^- 2 + ^-7 2 , I = - -J jf(a:',y')v' 2G o KRdx ' dl J or 1 = - i- 2 f JG KRV%x',y')dx'dy' - y { /(*'• ^°o«b - G o" R J/( a; '> y } ds \ - 2 4f(*,y) K the line integral being taken round the boundary of A. If this three-termed equivalent of the integral I be substituted in the series for <f>2 and 2 , each of these series may be subdivided into three, (f> 2 for instance into (ft T x cosh Kh sinh KZ [ (a k r WH(x' v'\dxdv k 2 /c/t(cosh 2kJi - I) j a series of the same general form as the original series, but at once more convergent, and of two orders higher in h ; z . . x "V 1 cosh kIi sinh kz ( f ., , i\ d n „ n y, d ., , , x I , ( u ) ^ K fe(cosh2^-i) J \f(*,y)^o«K-z^,f(*,y) \d», which corresponds to a strain local to the boundary of A ; ,... x , "V 1 cosh kIi sinh kz ■ , • , , -, . .-, (in) 2irf(x, y)Zj ~2 -T7 — ,i .; ; _ i \ > a senes which can be summed in the same way as (31), being in fact simply the first series of (31) with /3 = 0. The sum is therefore cosh k!l sinh kz 2irf(x , y) ■ coeHicient of k° in K 2 (sinh 2xh - 2k!i) We may now, by repetition of the same transformation, obtain a similar threefold THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 159 equivalent for the series (i), and continue the process as far as the continuity of the derivatives oif(x , y) will permit. We should thus obtain, in place of <p 2 , ... , , V, ,1 cosh kJi sinh kz f f ,„ , , , , (l) ( - )\f ( - i-s K/f(cosh 2jj r -i } j j G kR v "/(* , V )d* dy , a function of order h 2n+2 . (ii) A series of line-integrals which we need not write down, corresponding to a local perturbation at the edge of A, and giving the edge values of the relative part of <p up to terms in A 2 ". (iii) A series of n terms 27r ( C o/- C 2V 2 /+ cMf- + ( - )" _1 C2n-2V 2 "^y) , where c 2r is the coefficient of x ir in the ascending power expansion of (- ) 2/ • i 9 * _ o aV and is obviously a rational integral function of z and h of degree 2r + 2. If the function f(x , y) has its derivatives of every finite order continuous throughout the area A, the process can be carried to as high a value of n as we please, and we can thus obtain the values of (p 2 , $2 to an y required order in h. It should be noticed, however, that the series (iii) is not necessarily convergent when continued to infinity, as we may see by taking as an example f= cos ax, when the series would become 2w cos ax(c + c 2 a? + c 4 a 4 +•■•), which is the expansion (without the terms of negative degree) of n cosh ah sinh az -27rcos«u? — — , a^(sinh Aali — lah) and is therefore divergent if | 2a h \ > | ^ | , ^ being the complex root of sinh £ — £ = with smallest modulus. The form of the condition suggests that in ordinary cases the series will be convergent if h is small enough ; and when this is so, this part of <p 2 , 2 taken along with <p x , X will define an exact particular solution within A, giving the proper values of the surface tractions, and arranged in terms of ascending order in h. As a special case, the series will terminate if. for some finite value of n, ^/ 2n f^ , and in particular if f be a rational integral function of x, y. (It may be noted here that the solution for f= p m+ip cos moo might be obtained from the solution for /= Jmftp cos mw by expanding in powers of -/3, and equating coefficients of / 8 m+2 ^ in conditions and solution.) Looking back now to the ^ , 6 1 part of the solution, and having regard to (18), (29), (35), we see that we may write symbolically F= 2tt V - 4 /; v 2 F = 27TV- 2 /, and ^ =.2 fl -(c_ 4 v- 4 /-c_ 2 v- 2 /). The particular solution to any order in h is then given by cj> = 27r(c_ 4 v-y-c^v"y+c /-f 2 V 2 /+ — ) where c 2r is the coefficient of K 2r in the expansion of ( - ) . °° s , ^ s ™ Jf* , for negative as well as positive values of r ; or, as we may put it, this particular part of (p is given by expanding ( - ty jj^^kh*- 2Z1) » writin g ~ V 2 for K \ and operating on/(x , y). 160 MR JOHN DOUGALL ON AN ANALYTICAL THLOBY OF 15. Independent symbolical solution of the general problem. The form of the last result suggests a method of dealing with the problem from the beginning, which, though not easy to develop independently with thorough rigour, has the advantage of conciseness, and will therefore be useful in giving a rapid account both of the foregoing solution and of those to be obtained in the following pages. We begin by observing that (£ + iif + £) sinh -*■* v) ' sinh k2 CS + $ + K *) f{x ' y) = sinh kz(v 2 + K 2 )f(x , y). Hence sinh <zf(x , y) is a potential function, provided we regard k as an operator such that ie 2 = — v 2 - We may, if we please, take k = i^j , but it will not be necessary to interpret odd powers of the operator y. On this understanding, it is obvious from (12), (13) that we obtain a solution giving ZX = ZiJ — 0112= +/l ^l = on z= — h = j\x , y) on z = h , within the area A , by taking cosh k!i sinh k/i , \ 4 ^ = - K*(8inh2K/ 4 -2icM Sinh ""'J " ^( s inh2KA+2 K /^ COSh "*/ \ . (37) * 2 (sinh 2 K h - 2 K h) V K 2(sinh 2xh + 2k//) cosh kIi + 2kJi, sinh kIi sinh kIi + 2k1i cosh k!l ^ " ^( S inh2^.-2^) sin W + J^iiDhSrt + Si*) C ° ShKZ y ) Now, taking as a specimen the first term of iiu<p , we observe that the function of k cosh kIi ( - ) „, . , , — =-t\ sinh *z ' K 2 (sinh 2k/i - 2kA) vanishes at infinity round the path W A BE of § 5. Hence the function is represented by the sum of its polar elements. (If k 1 be a simple pole of the function, and if in the vicinity of this pole the function = A 1 /(k — k 1 ) + finite, then ^/(k — k^ is the polar element at this pole, and A x is the residue there. The point k = is a multiple pole, and the polar element there has the form A/k 4 + B/V 2 , these being the terms of negative degree in the expansion of the function near k = 0). Taking the elements belonging to zh Kj together we obtain cosh kIi sinh kz (-)-. /c 2 (sinh 2k1i - 2k1i.) A B . / 1 1 \ , -T- + — + A, + • K* K" \K — K-y K + Kj/ A B 2A.K, K K- K — K-, A B . cosh K,h sinh k,z + + ( ) l ' 1 2 2 the series extending over the poles with positive imaginary part. When we put k 2 = — y 2 , this part of 4/xc/> becomes Ay -4 /- By-y + 2j -w — Tin — Tv "T- -o • f, x J v • K k/j(cos1i 2k1i- I) y - + K- 2 •/ where k is no longer an operator, but simply a root of sinh 2kIi — 2kIi = 0. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 161 Now since one value of ( V' 2 + k 2 ) J JG kR/(z', y')dx'dy' = - 2tt/(x , y) , V + k- ZirJ J Similarly one value of v - !/' is — j J\og(R/2h)f(x, y')dxdy (38) and of v-*/ ~f i ( | K 2 log 2* - I R2)/(x', z/)^W Hence for the first part of 4p.<p we obtain + 2^ < - ) K/,(cosh 2k/, -1)]] G o«W(x , V )dx dy ) which agrees with our previous solution, the element of which is given in (20). Further, the results obtained at the end of § 14 clearly agree with what we should get by expanding the function of k in (37) in ascending power series and interpreting. As an example of this use of equations (37) we may find to a first approximation the value of zz at points not very close to the edge of A. From (5) (39) m = MW)-it**) + *&*+ i 2 /. „\ d 2 f A ,\ . d s t 2Z dJ (2 cosh kJi + 2k/(. sinh «//) sinh kz — 2kz cosh kJi cosh kz sinh 2k1i - 2kIi (2 sinh k/i + 2kIi cosh k/i) cosh kz - 2kz sinh k/i sinh kz f + sinh 2kJi + 2 k/i, Thus = (3h 2 -z 2 )zf/2h*+/. ? z ={(3h 2 -z 2 )z/lh*+l/2}f(x,y), and we verify at a glance that this gives the proper values at the faces. (40> 16. The problem of tangential faoe traction. Solution for an element of traction. We will now pass to the problem in which the given surface traction is tangential. Taking the direction of the traction parallel to the axis of x, we may take for conditions **■ = /0 c ,y) on z = + h | on z = - h > on z= ±h ) (41) zy = zz = According to the method explained in § 3, we begin with the function kJ (kR) in place of f(x , y), and determine potentials \k, 6, <fi giving ar-$ d 2 6 d 2 <j> d 3 <f> 1 _ _ r- r + 5 — r +7j+ 2z , ., ■ = 7 r- K J n KR on z = h dy dz dxdz dxdz asrdx 2/x ° _ dM_ d°-6 d-cf> d 3 <f> dxdz dydx dydz " dzkly ~ dz 2 dz 2 " dz 3 TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). on z= —h on z= +h on z= ±li (12> 26 162 MR JOHN DOUG ALL ON AN ANALYTICAL THEORY OF Since J,.kR = -— -. ., J,.kR — , t^J„«R, it is clear that these equations will all be k- (/.<•- ° k~ ay- ° ' 1 satisfied if we take where d\l/' a >W dcf>' d *' l 1 T P 7 ) — t = - s J n*K on z = li \ dz 2fj. k ° r = on z = — h j + ; + 2?: ,t = -K-- J„«R on z = A dz dz dz 2 d 2 ff d^' . d 3 <f>' — — - 4- 2z — — = dz 2 dz' 2 dz s on z s= — h ,. -- +h on (43) (44) (45) From (44), 4 ^'= -? s°inh2L J o kE '. or, separating the odd and even parts in 2, , 1 cosli k2 1 sinh kz T k^ smh kIi ° k-* cosh Kft u We also find easily Att<$> = sinh kIi J„kR sinh kz K 2 (sinh2/c/; -2k7i) u cosh kIi -97 — , o 3 , ,- J n KR cosh kz K 2 (sinh2K/i+ Zk/i) ° (46) 2«7i cosh k/; -- sinh kIi . 4u,P = 9 / • 1 n 7 n 7 \ J,,kR Sinh K2 ^ /<- (smh 2k/; - 2k/<.) ° 2k1i sinh k/7 - cosli k]i <LkR cosh /c2 K-(smh 2/c/i + 2k/i) ° Treating these expressions as in § 3, we find a solution for an element of X-traction at (x', y', h) of Stt/ul units given by (43) with 3inh kz f: k 2 cosh /</(' " cosh kz J„kR + e-* h - \d-K 1 e-* h / 1 r ( cosh kz T ^ 1 e-" fl / 1 1 1 \ ) J „ ( k- sinh km ° ni k/i V 2 6 4 // r" f • sinh kA sinh kz 3z e - *' 8 / 3 1 \) * = j i ~ K^mT2^-2WTy'» KR + « + ^< " ~2 Wz ~ J hz ) J dli r ( cosh kK cosh kz 1 e- K ''/ 1 1 \ ) / „ j K-(smh 2ic/i + 2k//) ° i^/t. &k/i \ 2 3 / j ['■ ( (2i<h cosh k// - sinh k/i) sinh «.-; 3z e- K, V 3 19 x i °- j n \ K-'(s>nli 2kA;- 2,<h) J " kR - 4^P ■" 8*W - 2 R " + h h Z ) ) dK K 2 (*n\\i2Kh-2Kh) u o R " 4k% 2 8/c/i'V 2"'^^ r- r (2«A sinh k// - cosh k//)cos1i kz t ^ 1 e~ Kh / „ 1 „ 11, A) + j„ 1 ^(sinh2^ + 2 i ) J «" R + 4^ + 8^( 22 - 2 E2 ~ T 7i2 j } ^ (47) These expressions may be transformed by the method of § 5, and a slight inspection of the relations between (16), (17) and (20), (21) will enable us to write down the (49) THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 163 results at once. These are, if we separate the parts odd and even in z, *»' = ?<- 2 )?S^S »(« E ) + ^x} .... («) where k is a positive imaginary root of cosh ah, y sinh *7i, sinh kz 3 / 1 \ 1 ^o = 4 j(_) ^(cosh2K/ i -l) G « KR - M" X " 6* V 2 XJ-40 Z V 2 X Z(2kA cosh k7i- sinh /c7;)sinh kz . _, 3/ 1 „■,«, n \ 1 n ^( g osh2kfe-l) - G o( kR ) + «<* - 6*Vx-27^v 2 x) + 4o*V-X + zV 2 X where k is a zero of sinh 2di - 2<h , with positive imaginary part. where k is a positive imaginary root of sinh kK. y cosh *cA cosh «z 1 / 1 \ h *' = 4*< " >«%(coah 2«* + 1) G ° kR ~ 47il X ~ 2 z " v *,) + 24™ •^ (2/c7i sinh k7i- cosh k7?) cosh kz 1 / 1 \ 117i '• = 4 K%(cosh2KA + l) G o(^)-^X - 2 z2 ^ 2 xJ--24 V 2 X where k is a zero of sinh 2<h + 2k]i , with positive imaginary part. The solution is defined by these equations with + = *£ e = ^-,4>=^L. dy dx dx 17. Composition of the solution. On examining the composition of the solution, we observe in the decaying parts of (p , , solutions of the class already obtained in (22), (25), and in the corresponding part of \^ , solutions of the type •A = sin(2» + l)gF(a;, y) (51) ^ = cos— F(x, y) , d\p _. each giving -% = on z = + 7i , and therefore zero tractions at the surface. As for the permanent terms, they may be arranged in the following groups, in each of which the surface stresses vanish. zi 1 c7 9 1(54 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF These are of the types (23), 24). (iii) d - dy 6 - ^ v "* This gives, by (4), since y 4 x = 0, ( P W = - *dxdy V X d , and may therefore be considered as of the type (24). It is important to note, how- ever, both here and in the cases immediately following, that the transformation on the value of u will not hold after the elementary solution has been integrated for the pur- poses of the general problem in (41). (iv). * h dy\ K 1 J»„S T ^V~X Id/ 1 ., , U dx\ x ~ Y zvx ill dxV (,- ¥^ Now F being any function of x, y, satisfying y 4 F = 0, the solution leads to d?F dy 2 d*¥ .3 - a „ d 2 + <" +1 )^ +i i- v iv^ ?o and z./- = ~y = zz = (y) <A dy h d dx dy h d „ which may be further decomposed into U dx* x > Uh d_ 24 dx V 2 X of the form (26), and the displacements corresponding to which vanish. h_ d 2 (52) (53) THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 165 18. General solution. Comparison with the solution for normal traction. The solution of the general problem of (41) may be found by multiplying the expressions for >//, f , </>' in (48) .... (51) by —f(af,y') and integrating term by term over the area A within which f is finite. As in the case of the problem of given normal traction, term by term differentiations of the resulting series will be legitimate just so far as the derivatives are required for the calculation of displacements and stresses. In order to see this, it should be noticed that, while an extra differentiation as to x or y will be required in virtue of (43), the series for \f/, &, cj>' have general terms of one order higher in 1/k as compared with those of (20), (21). One effect, however, of this additional differentiation will be to increase the relative importance at the edge of the area A of that part of the displacement and stress which arises from the local perturbation, such displacement being of order h, and stress of order zero, as in the former problem, whereas the displacement and stress as a whole is of higher order in h than before. The functions \|/, 6', (p' being symmetrical about the axis R = 0, it is clear that the solution for an element of traction of 8717/ units at (x', y f , h) parallel to the axis of y is given by *- -ft', 6 = f,<j, = d jL (54) ax ay ay withx//, 6', 4/ as in (48) . . . . (51). It will be seen presently that surface traction may be regarded as a special case of force applied in the body of the plate. We may therefore postpone any more extended development of the above solution, and in particular any more explicit comparison of the results with those of the accepted approximate theory of thin plates, until we have obtained the solutions of the problems relative to sources of strain situated in the interior of the solid. 19. Normal force applied at a single internal point. Solution in definite integrals. We take first the case of a single force, say for convenience of 4-^^(0 + 1) units, applied at (x 1 , y', z') parallel to Oz , the faces of the plate being free from stress. Referring to (6), we see that the conditions of the problem may be taken to be (i) « = ( 2 '-^ (IX v = (z'r-zfl +V . (55) dy (ii) U, V, W, along with their derivatives as to (x , y , z) of the first order, are finite and continuous at every point of the solid at a finite distance, and have derivatives of + u +v + ar~ 1 + w ; 166 MR JOHN OOUGALL ON AN ANALYTICAL THEORY OF the second order satisfying /*V 2 U + (A + p) d * = ; ^v 2 V + (A + /x)^ = ; /, V 2 W + (A + ^ = j (fx ay dz where . _ dV dV rfW rZx rty dz (iii) zx = zy = 7z = on « = ± A, It is clear that these conditions do not completely define the solution, seeing that no condition to be satisfied at infinity is mentioned. But instead of laying down any such condition at infinity, it is simpler in the first instance to be content with any solution fulfilling (i), (ii), and (iii). The most general solution can then be obtained without difficulty, and with this before us, conditions at infinity can be discussed to much greater advantage than at present. The problem is solved when U, Y, W are found in the form (4), so as to give the same tractions on z=dzA as those due to (6), but reversed. These reversed tractions, as follows very readily from (5), (8), are given by 1 - a -i , / '\ dr' 1 ^ T +{Z - Z) Hz- zx _ _ d %J~ dx 5/_ . d 2/x dy Now when z > z', 1 + adr- 1 . _ kcPt-i 2/jL 2 dz ^ Z) dz 2 e-^-^JoKRd/c J dr but when z < z', J C 2,.-l f dz- J o dz* ^-^JokIWk 7V dz J o ( ^Z1 1 = r K *e«*-^J K-RdK ..... (56) dz' J o Hence if U, V, W be defined as in (4), the function \|/- is not required, and the conditions to be satisfied by 6 , cf> are, if in the first instance we take integrands instead of integrals, dz dz dz ! t 2 ' ) = i l ~ - - K (h + z) i e-^+ z ')J KR, on «= - h = { - Lt?K - K%h + z') i e- «(*+^J kR, onz=-/t • . (57) ( <4 ) THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 167 Assuming (j> = A sinh kz + B cosh kz | 6 = C sinh kz + D cosh kz / ° J/,kR (57) give four equations to determine A, B, C, D. By addition and subtraction these are resolved into two equations for A, C, and other two for B, D. Thus we find , sinh kzJ„kR f - • i > , i / , 9 i « , \ ) <t> = —7- . . „ , " . , { kz sinh kz - if cosh kz (e' iKn + a + 2k//,) } K(sinh 2kJi - 2k//,) ( ) c osh kzJ kR ( kz , cQgh ^, , ginh KZ '/ e _ 2Kfi _ a _ 2k/ . I K (sinh 2k//, + 2k//) I ; \ j = f" hKZ «V K J _ KZ ' s i n h K g'( e - 2«/> + 2/c70 + cosh kz{— e ' *«* + — + a/c^ + 2k% 2 ^ I K(sinh 2k// - 2k//,) I \ 2 2 J ) cosh kzJ,,kR f , h kz ,, _ 2Kh _ 2k] , ginh J _ a_ - 2Kh + }_ + j + 2k2 ^ \ \ K(sinh2K/i + 2K//) I v ' V 2 2 J) . (58) If these expressions could be integrated with respect to * from to oo , the balancing displacements U, V, W of (55) would be determined. Near the upper limit the functions converge to zero exponentialwise, since both z and z' lie between — h and + h. But for k = both functions are infinite, and their expansions in ascending powers of « contain terms of negative degree which must be removed after the manner of S 3. The integrals are then convergent, but a further modification of a different sort is 1 necessary before they can be transformed into series as in §§ 5, 16. The possibility of this transformation in the former cases was intimately related to the fact that the functions in (13), (46) were odd in k, which the functions in (58) obviously are not. However, when the odd and even parts are separated, the latter are found to have a very simple form, free from the denominators sinh 2k1i dz 2*A, for we find <j> = -—sinh k{z - ;/) J kR 2k sinh /czJ kK ( / • i / i / i c 7 x i - ) + . . . -, , ' ,. < kz sinh kz - A (a + cosh 2k//) cosh kz > K(smh 2k1i - 2k//) I z x ' J , COsIikZcLkR f, ,/i/ in, xi -I + , . , n = — -— ; . < kz cosh kz + A (cosh 2k// - a) smh kz > K(smh 2 k//, + 2 k//) I i v ' J = - \ ~ sinh k(z - z) + z cosh k(z - z) \ J kR + , ■ ■, r.~T " ~ -, l ~ kz' cosh 2/c/t sinh kz + ( — cosh 2k// + — + 2k"//, 2 )cosh kz > K(smh 2k//, - 2k1i) I \ 2 2 / J + / • i — > \ " K n , . i kz' cosh 2/c/i cosh kz' + ( - — cosh 2k// + — - + 2k'-//' 2 )cosh kz' > K(smh 2 k//. + 2 k//,) I V 2 2 / j • - • (59) The even terms in «■ can be eliminated from these expressions by including the 168 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF values of (p , 6 which d« fine the source, given in (8). For these are <£= i. j' | e -^-Oj oK R _ e -" h \—, if z>z' = - i-f" | e *(z-')J oK R - e-^ 1 — , if .:<./. 61 = - — I { e " « 2 " *") J kK - e " «* i ! - + ::' j e " "< z " *>J kR^, if 2 > 2' 2 .' ip ( ) K JO = \\ i r*r.-z),) oK \i _ e-Kft I ' 7/< + ./ I %^-OJ oK RcZk, if e<z'. (60) These will obviously be reproduced, after preparation by removal of terms of negative degree and integration, from «■/,= —e-«( z - z )J KU, if z>z 2k = - J_e«( z - 2 ')J KR, if z<z'. 2/c _5L + -' y-^-OJ oK K, if. -j>;.' A +z ' W-*')J>R, if z<z'. When these last terms are taken in, the first lines of <p, in (59) become <t>= ± — cosh k(z— z')J kR '2 k 6 = + ■£- cosh k(z - z') J kR + z sinh k (z - z') J kR , the upper or lower sign being taken in the ambiguities according as z > or < z'. Hence, when the source is taken in, the following are the unprepared and unintegrated forms of (p , : — <j> = ± — - cosh k(: - z')J k-R 2k sinh kz.J„kR f , . , , ,, 1 o ; \ i > ) + — — ■ ; < kz sinh k.v - Ma + cosh 2k/i,) cosh kz > k(siii1i 2k/j - 2#cft) I J + —-.,- - ,- uK . < kz' cosh kz + i(cosh 2k1i - a) sinh kz' \ k(siiui 2k1i + 2/cA) I J 6 = + — cosh k(z — z'). I kR + z' sinh k(z - z')J kR 2k + / • t « t ° ft , v ^ - K2 ' cosn 2k ^ sinh kz' + ( ^ cosh 2k/j + - + 2k 2 /< 2 jcosh kz } K(smh 2k/i - 2/c/i) ( \ 2 2 / J + .°. S , jf ,°, a , . \ kz cosh 2kU cosh kz' + ( - ^cosh 2kJi + - + 2k 2 A 2 \wh. kz \ k(sdi1i 2kIi + 2kIi) t \ 2 2 / J (61) In (61) the terms of negative degree in k are of the forms: — in (p , H//c 3 + K/fc; in 6, L/k 3 + M/V ; and these terms, as in § 3, give zx = zv/ = zz = at the faces of the plate. (62) THE EQUILIBRIUM OP AN ISOTROPIC ELASTIC PLATE. 169 Hence if from <p in (61) we subtract H/^ + Ke-*"/* , and from 6, L/^ + Me"^, the resulting expressions, integrated with respect to k from to oo , will define a solution of the problem stated at the beginning of this article. 20. Normal force applied at a single internal point. Solution in series. To the integrals thus obtained we can apply the transformation of § 5, but one remark should be made. From the synthesis which gave (61), it is sufficiently obvious, in view of the forms in (58) and (60), that the expressions of (61), with Gt kR sub- stituted for J (kR), vanish effectively at infinity in the first quadrant of the k plane ; that they similarly vanish in the second quadrant follows at once from the fact that the functions of (61) are odd functions of k. A glance at the relation between (16), (17) and (20), (21) will again save us the necessity of writing down the details. Thus, let the values of H , K , L , M , when R is put equal to zero, be denoted by H , K , L , M . Then the persistent part of the transformed solution is given by 4> = H oX (R) - K oV 2 x(E) I 6 = L oX (R) - M oV 2 x(R) ] ' The decaying part is given by d, = 2^--,-,® — , „ ., — -,-\ \ kz sinh kz - 7r(a + cosh 2/</t)cosh k?J \ ^ K K/i(cosli 2k/i - 1 ) [ 2 J = y\ 7 T ,° K , S "Y K '\ \ - kz cosh 2k1i sinh kz + ( ^ cosk 2/c/i + ^ + 2/< 2 /t 2 jcosh kz } > . (63) , k/i(cos1i 'lull - 1) I \2 A J ) ; = y, ~rr- , - -, — Vs \ kz sinh kz - - (a + cosh 2k/i)cos1i kz I ( - cosh 2kIi) * Kh (cosh 2k/i - 1 ) ( 2 X J v ' where k is a zero of sinh 2/c/a — 2kJi , with positive imaginary part ; with <A = 2li -t /°" i o t K \\ \ kz cosh kz + „ (cosh 2k/i - a)sinh kz \ ^ K K«.(cosh 2k/i+ 1) ( 2 X ) = .Zj , ,° K . n , K ~- { kz cosh kz + ^ (cosh 2k1i - a)sinh kz \ cosh 2/Ji k /c/i(cosh 2kIi+ 1) I 2 X J where k is a zero of sinh 2kJi + 2k1i , with positive imaginary part. When the values of H , K , L , M are obtained from (61), it will be found that (62) may be decomposed as follows : — (ii) % I - %\ } each multiplied by J-^ - tf ) + 3 i|^( J W - J *) (iii) 4> = "-^v 2 x = ^ (iv) <£ = + iv 2 x> 0=±£<*v 2 x> with upper or lower signs, as z is greater or less than z'. (65) TEANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 27 (64) 170 MK JOHN DOUOALL ON AN ANALYTICAL THEORY OF There being no discontinuity of displacement or strain at the plane 2 = 2', except at the point where the force is applied, we are prepared to find that (iv) give no displace- ment at all if R> 0. So long, however, as we keep to the specification of the strain by the <p , 6 functions, it is convenient to retain the terms in (iv). By so doing, we of course make the <p of the space above the plane z = z' and the (p of the space below that plane two distinct potential functions, but we preserve the non-singular character of each of these functions at the axis E = . If we take the limit of the above solution for z' = h, which obviously may be done by putting z' = h in each term and using the lower signs in (iv), we obtain simply the solution of (20), (21) multiplied by ^(a+1). Since the present solution is for a force of 47r / u(a + 1) units, and the other for an element of traction of 8^^ units, it follows that a unit element of traction may be regarded as simply the limiting case of a unit force, the point of application of which approaches indefinitely near the surface. 21. Solution of a special problem of internal areal normal force. When the displacements due to a unit Z force at (as', y', z'), with the surface free, are known, the corresponding displacements for a body distribution of force, of amount Z(x', y', z') per unit volume at (x f , y', z f ), can be found by multiplying by Z(x r , y', z')dx' d'f dz' and integrating through the space in which Z is finite. Certain peculiarities in the form of the solution given in ^ 20 make it convenient to take the integration with respect to z' last, or, as comes to the same thing, to begin by con- sidering the solution for an areal distribution of force on the plane z = z 1 ', of magnitude Z(x', y r , z') per unit area. We take first a special problem analogous to that worked out in § 10, and suppose the Z force to be distributed over the area of a circle of radius a in the plane z = z\ with centre on O2 , the intensity per unit area being 47^(0 + l)J m (3p cos moo. It will be sufficient to attend to the value of <p, for when that is known, the corresponding value of 6 can be written down at once. The series deduced by integration from (63), (64), say <p 2 , fall naturally into two parts as in § 10, viz., (i) series defining a local perturbation at the cylinder p = a, 4, = ^ Kftfcosh -2k}i -1) \ KZ ' Sillh KZ ~ Ty ( a + cosh 2k7(.)cos1i kz' | . ^" k2 P« (where sinh 2kU - 2kJi = 0) + 2-L-.- r .- -= ' , x \ kz' cosh kz' + - (cosh 2k1i - a)sinh kz } • na "* n P K (where sinh 2i<h + 2k1i = 0) K kA(cos1i 2k/i+ 1) | 2 ' ) ft 1 - k 2 with P„ = J„,k/3 cos m«>(i<aG J kccJ ,„/3a — G m Kaj3aJ „,' fia) , if p<a = G,„Kp cos 7)iw(ko,J ,„' kci3 Vl f$a — J,„Ka • (3aJ„'{2a) , if p>a (66) (ii) When /><«,, series which can be summed in finite terms, <f)= 2-7T-J .„,/?/) cos too y. smh KZ — -, { kz sinh kz -\{a. + cosh 2/Ji)cosh kz \ (where sinh 2kJi - 2 K h = 0) « (p 2 - k 2 )k/i(cos1i 2«/i - 1) (2 J 4. zL - '—7 ' h " ^ \ kz cosh kz' + - (cosh 2k1i - a)sinh kz' } (where sinh 2k// + 2k// = 0) _ + «(yS 2 - k 2 )k//(cos1, 2,<h + 1)1 2 ' J v (G7) THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 171 In order to sum these last series, consider the function of k } cosh k(z — z) / . \ + o , n« w~ ( ± according as j> or <z ) 2k{[J~-k~) \ ° J sinh kz ( , . , 1 \ sinh kz (" , , i + (F-K>) K (smh2 K h-2Kh) \ KZ smh KZ - K« + C03h 2 ^')cosh kz' J. cosh kz ( , , . , ) + 77w g\ / • t. n / n~~ iv •! KZ cosh K2 + A(cosn 2kA — a)smh kz > . . . (68) (p~ - K-)K(simi 2k/i, + 2k/i) \ - v ' j v ' Looking back at (61) we assure ourselves that this function vanishes at infinity in such a way as to make the sum of its residues zero. Also, since the function is odd in >c, the residues at k= ± k 1 are equal, and therefore the sum of the residues at the zeroes of sinh 2kJi ± 2kI% is simply the coefficient of 27rJ m /3 j o cos mu> in (67). The sum of the residues at k = ± /3 is cosh f$(z - z) + w~ sinh f3z MsinhVph- 2(3h) { ^ sinh ^ ~~ ~^ a + C08h W cosh P z ' cosh /?z 2 (sinh 2(3h + -2j3h) i /3z cosh flz + £(cosh 2/3A - a)sinh /3z' I . . (69) A P If this last expression near /3 = be of the form ^ + -5 + . . . , the residue at k = q of (68) is simply - j^ - -p . Hence the coefficient of 2TrJ m fipcosmw in (67) is simply (69) with sign changed and the terms of negative degree in /3 subtracted. These terms of negative degree, just as in § 12, are added on again when we take in the part of the solution coming from (65), which is obtained by writing F for x in (65) where F = II x(R)J m /3p' cos niw p dp' dm. 2tt — "04 Jm/^P cos mm + Fi • (Introd. (/).) The term ~^ m fip cos mw being taken in for the purpose just mentioned, we are left with F : instead of x in (65). Since y 4 Fi = , these equations now define a combination of deformations of the persistent or permanent type, under no body force and no surface traction. The solution therefore resolves itself into (i) this free deformation of the permanent mode ; (ii) a local perturbation ; (iii) a particular solution giving the proper discontinuity of stress correspond- ing to the applied areal force. 172 MR JOHN DOUOALL ON AN ANALYTICAL THEORY OF The particular solution is - 2 cosh /?(z - z) \ ± 2/3 ^ = 27rJ„,/3/)cosm„» J sinh/fe Mjd pr= mP z ' smh * " > a + cosh m cosh *'> + ^sinhtA 2^) ( ^' cosh ^ + * c0 ^~ 2 /^ sinh & (70) with a corresponding expression for d, obtainable from (61) by changing k into /3 and then replacing J /3B by g '2*3 J* p cosmw . It is easy to verify that this is actually a particular solution. Consider in the first place the analogous forms of <p , in (61), and for greater generality, suppose J kR replaced by f(x, y) where ( V 2 + * 2 )f= . Then, from the method by which (61) were found, it is obvious that they give no stress across the planes z — ± h . Let us examine the effect of the discontinuity in the forms of <p , at the plane z = z , on the displacements and stresses as given in (4), (5). If we take simply = g- cosh k(z - z')f — a cosh k(z - z')j—z sinh k(z - z')J then we find at z = z\ u — v = w = ;.r = nj = S = -//(a + l)*/'. Thus with the complete expression (61), the displacements are continuous, as also the stresses z7, zy, but ^ (z = z + ) exceeds » (2 = «' - ) by - 2m(« + 1)*/*. We thus see that in (70) the corresponding discontinuity in 7z will be — 4x/<(a + 1) J m {3p cos ma> . This continuity of displacement, and discontinuity in zz, are precisely as demanded by the conditions of equilibrium of the plate. If we take (61) with J kR unaltered, prepare them for integration as in § 19, multiply by e' Ke and integrate with respect to k from to 00 , the discontinuity in zz at z = z' will become -2fi(a+l) e-« K,J KRd,K . If further we multiply this by Z(x', y', z')dx' dy' and integrate with respect to x', y', and then take the limit for e = 0, the discontinuity becomes, in virtue of (11), -brfi(a+l)Z(x,y,z').. We have thus a proof of the solution for an areal distribution of Z force, independent of the infinite solid solution (6), which might itself be found from the beginning by this method. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 173 22. The general problem of internal normal force. Approximate forms of displacements and stresses. The developments given in §§ 14, 15 may obviously be applied in the present case also. Thus if in (61) we divide by kJ kR, expand in ascending powers of k, put — V 2 , that T-2 + j— 2 ) f° r K% i anf l operate on Z (x, y, z'), we obtain a form of solution which, with the interpretation of V ~ 4 Z and V~ 2 Z given in (38), is simply the foregoing general solution for areal force of intensity 2m(«+ 1)Z , arranged in terms of ascending order in A 2 . The solution in this form fails if at (x, y, z), Z or any of its successive derivatives become discontinuous, but it has been shown in § 14 how the local perturbation in the neighbourhood of any surface of discontinuity may be calculated. For the case when Z vanishes outside an area A, the principal part of the perturba- tion at the edge of A, when h is small, is found by substituting for G kR in (63), (64), ^ j { Z(Y, y') ^ G kR - G (*R)^ %&> //) J ds , where differentiations and integrations have reference to the accented coordinates. Since the solution for the case when there are any finite number of surfaces at which Z or its derivatives become discontinuous can be found from this elementary case by simple summation, we see that discontinuity in the force itself gives rise to values of <j>, 6 in the perturbation terms of order h 2 at the surface, discontinuity in j- to terms of order h s if Z itself is continuous. The next term is of order A 4 and depends on discontinuity of V 2 Z , that is, of the second derivatives of Z, and so on. The symbolical solution for Z force distributed on the plane z = z with intensity 2/u(a + l)Z(cc, y, z') per unit area at (x, y, z / ) is given by <f> = ± ^ cos1 ' K ( :: ~ "-') sinh kz ( , . , , \ 7\ [kz sinh k,: - \a + cosh '2k/i cosh kz \ ^ K 2 (sinh 2k7i - 2k/;) cosh kz ( . , ,\ + ~ K \smh 2 K h + 2Kh \ KZ Cosh KZ + i cosh 2Kh - a smh KZ ) a _ Z + ^— 5 cosh k(z - z) + - sinh k(z — z ) o/ • -L -» i u \ I - kz' cosh '2k1i sinh kz + ~ cosh 2kIl + h + 2k 2 Ii 2 cosh kz) « z (smh 2i<h - 2kIi \ 2 * J cosh kz + - 9/ ■ r -. 7 , o i.x ( kz cosh 2kK cosh kz + \ - - cosh 2kIi+ 2k-//.- sinh kz ) K 2 (sinh 2k/i + 2k1i) \ 2 / with k 2 =- v 2 , operating on Z(x, y, z') . . . . (71) The approximate solution is obtained by expanding in ascending powers of k 2 . By retaining only the terms of negative degree in k 2 , each of the displacements will be 17 1 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF given to a second approximation, and each of the stresses, except ze, to a first approxi- mation. The result is obviously the same as that found by integrating the permanent terms (65) of the original source solution. If we write T? = jfz(x',y',z')x(R)dx'dy', then the displacements for Z force of intensity Z(x, y, z') per unit area on z = z' are d , It. 3 / .\t-i „,, /a+5,,a — 3/o +1 ,, a — 3 79/ - (a + 1),~F + v "* -s- v + -. - 2 -z - -Z- hh + —— hh dy d SJhrfA 8 | v ' v I 6 2 5 3 ^ { (. + I)F + <*£?*+ ^-W + if 1 V) | 32^ ' ; ■' ' : ( ' The corresponding results for a volume distribution of force, Z(x, y, z) per unit volume, are found by integrating these with respect to z' from — h to + h. In order to calculate the stress zH from displacements, we should need the value of w to a third approximation. It is therefore easier to find zz directly from (71) and (5). On dividing by 2n(a + 1), we find, corresponding to (72), 37i 2 z - z l £=*iZ + — mT Z ■ ( 73 ) When the force is Z(x, y, z) per unit volume, this leads to zz= 4 p -J Z(x,y,z)dz --J Z(x,y,z)dz+^JZ(x,y,z)dz . (74) We can now find the stresses xx, xy, yy to a second approximation. For il x ihj dz -- v du , .dv ,v , .cto " = A dx + % +{X+2fl hz> whence, eliminating c ^° , ° «z ~ 4:ii(X. + u)fdu , cfoA A _ as« = -V-: :V I — + °"— ) + ; ,- zz, X + 2/j \dx dyj X + 2 and similarly Also * y==t \dy + Tx)- We have now only to put in the values of u, v from (72) and the value of zz from (73). THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 175 3 If we denote ,Ja + l)F, the principal term in w in (72), by W, we have 'o+l)Z 16/*/; 3 X + 2/x z 8/x/t 3 A + /x or Cv 4 W=Z, where C = | A i/i 3 (A. + / a/(A+ 2/x) . . . (75) In this notation, to a first approximation ~ = _ 3C_ f d 2 W d 2 W °° x 2hf\ dx> d~y* 3C / rPW <fW 2hf\ dip If yy= -rf°" tzt 4 3C M W <PW ,„. Again from (72), Vcfe (&k/ 32irfj.h 3 = i C ^-^^V 2 W . (77) 23. Normal force a function of z onfo/. It may be useful to put down here the next term in the development of w, of which the two principal terms are given in (72). This is 327rfxh V 4 F 24 l ^ ; + 4(a+l) + V 10 a + l/ l ' \8a+3 5 + 3-52.7 / (3 — a) 2 , o , 3a + 3 (78) ±(l-a)(z-z') Z(z,y,B') In this, of course, V 4 F = 2ttZ(x, y, r/). The terms which have to be added to (72), (78) in order to give the complete particular values of u, v, w, all contain x, y derivatives of V 4 F or Z. Hence, if Z(x, y, z') is a function of z' alone, (72) and (78) give a complete particular solution of the problem. Further, Z may have one constant value in one region of the plane z = z', and another constant value in another region of that plane. (72), (78) will still give a particular solution in each of those regions taken separately, or rather in the cylindrical spaces of which these regions are sections, but it ought to be carefully noticed that it is not in general an exact solution when the two regions are considered together as part of one body. The point of failure is, it need scarcely be said, the condition of synexis ; 176 MR JOHN DOUG ALL ON AN ANALYTICAL THEORY OF the two particular solutions do not fit, that is, they do not give the same values for displacements ami strains on the two sides of the cylindrical surface or surfaces of discontinuity. On the other hand, the supplementary terms required in order to make the solution synectic belong to what we have called the decaying type. They give rise to displace- ments and strains of infinitely high order, if we may so speak, in the small quantity h, except very near the surfaces of discontinuity. This being so, we need not be surprised to find that the solution (72), (78) is not necessarily the simplest particular solution in any one region within which Z is continuously constant. Thus, for example, if we pick out the terms which contain z' 2 as a factor, we find displacements proportional to *V 2 F w = a_ dx dy v 2F +v 4 f(^^ + c^ which belong to the type (23), and contribute nothing to body force or face tractions. These terms might therefore be omitted in any problem where the condition of synexis is irrelevant, and in particular when the object is merely to obtain a particular solution for body force and face traction in a problem relating to a finite solid. 24. Internal force 'parallel to the faces. We will now go on to consider the problem of force applied to the body in a direction parallel to the faces of the plate. A force of 47r / «(a + 1 ) units applied at (x f , y r , z') in the direction of 0.x gives in an infinite solid displacements defined, according to (9), by >A = a+l d 2 dy dz~ 2 a d d-' 2 r~ l ,d d~ : r~ l _ + z — 2 dx dz"' 1 dx dz~ x 1 d d- 2 r~ l 2 dx dz 2 Hence the tractions which such a force produces on z = ± h will be neutralised by a system \f/- , 9 , </> for which d\jr _ d fa. + 1 d~h'~ l dz dy\ 2 dz~ l d A + !*£ + 2z d2< t = d ( a + 1 d ~ lr ~ l dz dz dz 2 dx\ 2 ~dz zr + iP6_d 2 cf> :) d 3 <f>_ d/a-l . dr- x \ dz* dz 2 dz* riA~2~ r +z ~ z ~dJ) -on z= ±h . THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. These conditions are satisfied if and We may take </' dy' dz dz dip _ a + 1 d dz dz dff d<b' dU' a+1 d~ l r~ l , d 2 6' d*d>' d s d>' a-1 , dz 2 cfe 2 + Zz dz* ~ 2 r + Z - 2 <2r' tfz j-y- 1 dz~ 2 dry dz = P ' j eT^-^J^R _ 1 + K ( z - jQfl-i* J ™ _^=^( + ) je+^-OJ^R--- e+^-m k R^k 177 (79) upper or lower signs being taken, as all along, according as z > z' or z < z' . We therefore determine provisional values of \//, 0', <£', such that d\l/' a + 1 _ , , „ d 2 6' d 2 d>' d z <h' fa - 1 A , W - W +2z l§- = {-2- + KZ -* )^(^)J„kE These provisional values are easily found to be ^r' = C ^- r COSh/c(2-2') a+1 2k 2 cosh xh a+1 2/c 2 sinh kJi sinh kJi sinh kz' sinh «2 V J kR cosh kJi cosh k2' cosh kz $ = — 2 cosh k(z - z) sinh k2 -—- < - kz' cosh kz' + — (cosh 2kA - a) sinh kz' > IkIi) I 2 J K 2 (sinh 2k/i - 2k/j) cosh kz ( i I -—: — — - < kz sinh kz' + — (cosh 2k7i + a) cosh kz } 2kJi + 2kA) I 2 ) K 2 (sinh : J kR 9' = -— 2 cosh k(z- z) - — sinh k(z-z) 2k' k + .. . ? r 7 KZ „ , , < kz' cosh 2k/i cosh kz' + — (a cosh 2k/i - 1 - 4k 2 7i 2 ) sinh kz > K 2 (sinh 2k7i - 2kJi) I 2 J „ . , „ , KZ 7T I kz cosh 2k7i sinh kz + — (a cosh 2k/i + 1 + 4k 2 /V-') cosh kz' > K"(smh 2K/i + 2K/i) ( 2 _L TRANS. ROV. SOC. EDIN., VOL. XLI. PART I. (NO. 8). JqkR (80) 28 17! MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF The source itself is similarly given by the temporary values r a+ 1 Ik k > c=F«(«-OJ kR ~ o '1 Thus, when the source is included, the provisional values of \|/-', &, (j)' are as in (80), but with the first lines altered, in i// to ± „ sinh k(z — z) 2k' „ <f>' „ ± -—. , sinh k(z - z) Ak „ & „ ± s-s sinh k(z-z) + — cosh k(z-z') 2k k I <LkR (81) 25. Solution of the problem of internal force parallel to the faces. From these expressions the solution in the form of definite integrals, and finally of series, is obtained as in the previous cases. After the explanations already given, it will be sufficient to write down the final results. For the transitory part of the solution , \p' = 2j (a + 1) — sinh kz sinh kzG kR , (k a pos. imag. root of cosh kK) k K ll - 2-i (a + 1)_ cosh kz cosh kzG kR , (k a pos. imag. root of sinh kTx) . _ 'V sinh kz G^kR K I = 2a same as previous line multiplied by ( - cosh 2kJi) 4> — 2-i „ ^"i^l^nf^L ) _ KZ ' gosh kZ ' _j_ (cosh 2k1i - a) sinh kz' \ k-/« (cosh 2kK- 1) I 2 j where k is a zero of sinh 2di — 2k1i, with pos. imag. part. With '=£<-), $= cosh KzG n /<R k 2 A (cosh 2k1x + ] ) < kz sinh kz' + — (cosh 2kK + a) cosh kz > 6' = 2-i same as previous line multiplied by cosh 2k1i K where «■ is a zero of sinh 2kJi + 2>ch, with pos. imag. part (82) We may recall the method of obtaining the permanent terms. Taking any one of the functions of (80), altered as in (81), we omit the factor J a:R,, and then find its expansion near k = to contain terms of negative degree in *, say A/k s + B/k. The THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 179 permanent part of this function is then Ax(R) - By 2 x(R)- We tnus nncl for tnis P art of the solution 3(o + 1) , / 1 3 2 \ . 3 (a + 5 , 3 } , , _ a + L v \. _ 1 / ,V , + y(2-2)V'X , 3(a+l) ,/ 1 32 o; , . \ 3 /a + 5, 3 j„, a+LjA + ^F^ H 2X " T 2 V " X - 27r2V " X ) " 4/?^ ' k ' Z ~ W h Z > VX (83) When z' is put equal to h in the above values of \f/, <//, 0' it will be found, with very little trouble, that they reduce to those of (48) .... (51), multiplied by |-(a+ 1). {Of. § 20.) As in § 20, the displacements due to the ambiguous terms in (83) are null if R > 0. But there is this difference in the present case, that they do not continue to vanish in the corresponding solution for an areal distribution of force onz = z'. If the intensity of the distribution is X(x, y, z') per unit area at (x, y), this solution is defined as in (79), \//, 0', <fi being obtained from (82), (83) by multiplying by and integrating over the area within which X is finite. When this is done we find that the ambiguous terms lead to u= +—(z-z')X •Ifx. v = w = (84) In verification, we observe that these displacements are continuous above and below the plane z — z', and that the corresponding stresses are also continuous with the excep- tion of zx, the value of which just below z = z' exceeds its value just above by X. The value of «e being TfX, we have for the contribution of (84) to the resultant J zx dz, yX(f dz- j\foWx . . • - • (85) with 180 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 26. Approximate values of the displacements. Lagrange s equation for flexure to a second approximation. The unambiguous terms in (83), as in (82), fall naturally into two classes, in the first of which >//, 6', ft are odd functions of z, while in the second they are even. Of the displacements derived from the first class, u and v are odd, and iv even in z, and the strain may be described as flcxural. In the other class u, v are even, and w odd in z, and the strain may be described as extensional. A force X at (x f , y', z') acting along with a parallel but oppositely directed force X at (x', y', — z') would give rise to flexural strain only ; equal and similarly directed X forces at these two points to extensional strain only. This follows at once from the fact that the terms of \J/, 6 r , <fi', which are odd in z, are also odd in '/, and vice versa. The distribution of force being X.(x, y, z') per unit area at (x, y) on z = z', let Y = jLjjX(z',y',z') x (R)dx'dy'. Then from the flexural part of (83), *"H5P'(*-M <£=-£V'-'F| . I4 _.,. ,, 3 2 /o + ll, v a + 5 ,A e= SV , F f each multiplied by 32 — K , • ^-jQ- *"* ~ 12 z 3 ) These lead to _ d dy I For Y force the same expressions hold if we take F = ^ M Y(*', y', 2') x(R)^*W- These formulae, with all of (72) but the last terms of u, v, and with the odd parts in z arising from the ambiguous terms, give to a second approximation the displace- ments of the flexural mode under any forces. The differential equation satisfied by w, the normal displacement of the mid plane, or value of w, for z equal to zero, is important in the history of the approximate theory. We can now write it down to a second approximation, namely, with C as in (75), cv^ = z + -(^ + ^) V dx dyJ (a- 19 h- 3-<xz' 2 \ 9 „ /.,., a + 5z' z \ /dX dY \ THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 181 This equation gives the result for an areal distribution X , Y , Z on z-=z' . For traction on z = ±h replace z' by ± h ; for a volume distribution X , Y , Z replace Z by i K _Z(x,y,it)dz' ,/dX dY\ , d f> ,„. ,.,, d (x and so on. 27. Extensionai strain. Differential equations of the 'principal mode. The unambiguous extensionai terms of (83) remain to be considered. Write E = 3 2 J jX(x',y',z')x(B,)dx 7 dy'. Then for an areal distribution X , these terms are $= - 4(E - £*VE) + 4(iz' 2 + ^ 2 )v 2 E 2 / a + 5 ,.. 5 '- -(E-^v'E) + ^^ + ^ + a+l/r J V 2 E 2 / a+5 ,, 5-a„ \ „_ +' = - (E - ! 2 VE) + --j ( - J-W + 6 /r J V 2 E The second parts of these expressions give ; = (a - 3) (J/ 2 - J ft 2 ) ^ 2 V 2 E + 4(z' 2 + §/r) V ' E J 2 w; = E The first parts give u d 2 E Q d 2 E , o /a -3 rP . g tZ 2 E , 2 ^ a ' «te dy dx dv d*E_ dy 2 dx 2 a - 3 tf 2 2 dxdy V 2 E + 4v 4 eV V 2 E d w = (3 -a) ^ V 2 E <ia; If now further we write K = ^-^ / j Y(tc', ?/', z') x (B,)dx'dy' the corresponding displacements for a distribution of Y force on z = z' can at once be written down from symmetry. The results for X and Y force combined cannot con- veniently be expressed in terms of one function, as in the case of the flexural mode, and the best plan is probably to put everything in terms of the principal values of u , v , namely, U= -(o + l) ^ 2 E „rf 2 E , . 1N d 2 K . a d 2 K] dx 2 dy 2 dx dy dx dy I y- -( a + i)^ 2E +8 <* 2 E _/ a+1) ^K _ 8 d 2 K\ dxdy dxdy dy 2 dx 2 J (88) 182 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF We have then X ^ fxh g-3 ,,/<*£■ i*y a + 1 ' \ flte ay (89) The ordinary approximate theory obtains differential equations to determine U , V . These are easily found by eliminating E and K in turn from (88). Thus dTJ dV _ _, ,-i\(d 2T£ + d 2 K \ dx dy \dx dy J dy dx \dy dx I and 1 d_(dV dV\ ,idfd]J _dV\ X ^ a+ 1 tfaA cfo <ivy / c?//\ (/?/ da / [6/j.fi _J_ d/dV dV\_ 1 d_fdV _dj) = Y a+ 1 ayl 'te ^/7/ / dx\ dy dx ' \QfihJ (90) The principal parts of the contribution of Z force to extensional displacements appear in (72), (78) . In the notation of those formulse dx ', = /v 2 F dy a+ 1 2V 4 F 327r/x/i with in addition, w = the odd part in z of the ambiguous term in (78). If these last values of u, v be included in the principal values U, V, then the right- hand members of (90) will become respectively 1 1 a — 3 ,dZ \ 1 / a — 3 ,dZ a+1 ax j , lbfjJi\ a+1 ay (91) 28. Approximate values of the stresses across a plane parallel to the faces. For any distribution of force parallel to the faces of the plate, the formulae of §§ 26, 27 give the terms of the two lowest orders in the values of u, v, and the term of lowest order in w* From these terms we can calculate all the stresses but S to a first approximation, and as in § 22, when the first term of 7z is known, we can find two terms of xx, xij and yy. This first term of zz we may get very easily from the symbolical form of the solution corresponding to (80), (81). Thus for areal force X £= ±W- ?: ) dX 3 ,„. , R dX 1 ., ., dX zV™-\*-)liZ -T h (* 2 + h-)- ~dx + 4/t 3 dx ih x dx (92) * There should be added from (84) the terms ,' = T(-->~')X/2 M) «=T(s-« , )Y/2/ tl '" = 0. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 183 From (84), (86), (89) for areal force X 3 ,,.. l d 2 sy- + hX + ^ z'(? - h*) ¥i? ^ J jX(x', y, z) y \(^)dx'dy' + ~X ' W ZV ~ h *)*r didyj j X ^'' ^ OVxCB)^'^ (93) "zr «zy "2 It may be verified that these give zero stress onz=± /i, and -v- + -p + -5- = 0. From the formulae we have given, it is of course merely a matter of the simplest algebra to calculate any of the stresses to whatever order of approximation may be required, but it may be worth while to remark here that the fundamental equations of equilibrium (1) may be used with great advantage in obtaining the principal results. If. for example, we know only the first terms of xx, xy, yy, the two first of these equations would give the first terms of zx, zy by a simple integration with respect to z, and then the last equation would give the first term of zz. Similarly, when the first two terms of xx, xy, yy are known (as above), we may find the first two terms of the other stresses. 29. Transmission of force to a distance. Expansions in polar coordinates. We have up to this point been considering mainly the particular solution to which our general source solutions lead for any given distribution of force ; or, as we may say, we have been investigating the effect of any given force system on that part of the solid to which the force is applied. But it is also of great interest to inquire what is the effect of this force at points of the solid remote from its region of application. It is obvious that we obtain a sufficient answer to this question by retaining only the permanent terms in the source solutions, those terms, namely, which are given in (65) and (83). For force applied only at points on a given normal to the plate, these formulae are all that we require. They show at a glance that the distant effect depends chiefly on resultant forces and couples, but not entirely, since z' and z n occur in the formulae for Z force, and z n , z' z in those for X force. When the force is not confined to a line, but is distributed over a finite volume of the solid, the result is obtained in more intelligible form if before integration the function x is suitably expanded so as to yield a series of solutions in which accented and unaccented coordinates are explicitly separated. The most convenient expansion of x is m terms of polar coordinates as given in (e) of the introductory section. Suppose, then, a single force applied at the point (x x , y x , z r ) or (p x , w 1 , %), the components of the force being X 1 , Y x , Z x , parallel to the rectangular axes, or P x , Q x , Z 1 parallel to radius vector, transverse, and axis of z. We have to find the displacements at (p ,w ,z) where we suppose jo > p L . For an X force, the value of ^ is -^- with \|/ given in (83) , the coefficient de- pending on the magnitude of X being for the moment suppressed. 184 Ml! JOHN DOUGALL ON AN ANALYTICAL THEORY OF This is the same as — ■$-, or ( — ) rate of variation of \J/ in the direction perpen- dicular to the force at its point of application. Similarly Hence for <£ = _ -X _ ( _ ) rate of variation of <f>' in direction of force. dx 1 cty' * = d Pl and for O, «P = ^Pi J \dfi_ p, do)j * = -~d^J We shall take separately the extensional and flexural parts of the solution. Also in the following u, v are the displacements along radius vector and transverse. I. Extensional terms. The following solutions occur. [1 - a. 9 + a, p 3 - a 9 _„ \ s (!) U = ( — r log ^ + -j-Z^p ) COS W -7-a 9 + a. p 3 — a „ _, 2 u= (V + ^ l0g 4 + M> = (3 — a) Zp~ x COS w (ii) Same as (i) with cos w changed into sin w, and sin w into - cos w (iii) When m>l, ; ^^2 n — l I COS mco 4(to-1) -mz'p .-{ 3-, m(a^_ m+1 + ^-a m ^ 2_ — m- -il 4(m-l) M> = (3 - a)zp""' COS TOW (iv) Same as (iii) with cos mw changed into sin mw, and sin tow into - cos tow. (y) U = p~ m ~ 1 COS TOW I v = p m x sin ???w (vi) ?/ = p m * sin tow v= — p m— 1 -r?i-l COS TOOJ } For the force with components F 1 ,Q 1 ,Z 1 , the coefficients of the above solutions are the following, in each case divided by 32-Trfj.h. (i) Pj cos Wj - Oj sin Wj = X x (ii) P x sin w x + AJ cos Wj = Yj (iii) Pj"'" 1 cos TOw 1 P 1 - P!'" -1 sin mw 1 1 (iv) Pi" 1-1 sin tow^ + p," 1-1 cos tow^, (v) { "^ffiffi+V " + (3 - -) (K - ^)-P™- 1 } oo- -^ + { 8(m+ 4 g; + ^ a ^ r +1 - (3 - a) (^ 2 - ^Wx"- 1 } -n mcoA + (a — 3)p 1 '"z 1 cos towjZj (vi) Same as (v) with cos mwj changed into sin mw x , and sin mwj into - cos jbu, THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 185 II. Flexural terms. The flexural solutions are of the form given in (23), or in polar coordinates d ■ (a + l)(zF - \z? V 2 F) 4 2 (1 2 3 _ m) y 2 F dp 1 d p db IV = (a + 1)(F- 1-22 y 2 F ) + 2(.; 2 - fc») V 2 F (94) (i) F= x (p) = Vlog^-ip (ii) F = i( P -2plog|-)cosa (iii) F = i^-2 P log 2 y sin a (iv> F =4^=rr (vi)F = log£ >- ?»■> 1 . (vii) F = p '" cos mo> \ (viii) F = /j - "' sin ??iw ' m>0 For the force with components P x , Q x , Z x the coefficients of the above solutions are the following, in each case divided by 4f tt/x/i 3 . (i) Zi (ii) - 2 X cos ojjPj + 2j sin WjQj + Z^ cos a>j = - X^ + ZjSBj (iii) - z x sin oj 1 P 1 - z l cos WjQj + Z 1 p 1 sin w 1 = - YjZj + Z^ (iv) -z^np^"^ 1 cos raojPj + ;j 1 ?»/3 1 '"~ 1 sin maj 1 fi 1 + pj m cos maijZj (v) Same as (iv) with cos ?ri(o 1 changed to sin mwj , and sin mwj to - cos jhgjj (vi) - \z lPl v x + { ^ + iv - &* + JL(% 2 - **) I Zj ( a + 1 J (viii) Same as (vii) with cos ma, changed to sin mo> x and sin mu> l to - cos vtu 1 . 30. Types of deformation conveying a given resultant stress. In these formulge we remark at once a striking relation between the forms of the displacements u, v, w in the various solutions, and the multipliers of P x , Q x , Z : in the coefficients of the solutions. In I. (iii), e.g., these multipliers are f) 1 '"" 1 cosma> 1 , — pj" 1 " 1 sin ?nco 1 , , which are simply the displacements of I. (v) with sign of m changed, and consequently suitable for space containing the origin. TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 29 186 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF Similarly in I. (v) the multipliers of P : , ^ x , Z l are displacements compounded of the types (iii), (v), with sign of m changed, and so on. The full explanation of this peculiarity will be given presently, when it will be seen that an independent verification of all the results may be obtained by means of the important principle known as Betti's Theorem. In the meantime we may examine the scheme of solutions from another very important point of view. With reference to any individual solution, the following questions are obviously of prime importance : — ( 1 ) What is the resultant stress transmitted ? (2) Is the whole potential energy of the part of the solid bounded internally by a given cylindrical surface, finite or infinite ? Now, in order to single out those solutions which convey a finite resultant stress across any cylinder (or other surface) surrounding the origin, we have merely to look at the table of coefficients. Thus, for instance, I. (i) appears with coefficient X 1 /32Triu.h, from which we may infer (as verified below) that this solution conveys a stress with resultant a force of S'l-n-^h units parallel to the axis of x, and passing through the origin. In this way we find that the six solutions, corresponding to the six elements which specify the resultant of a force system, are I. (i), (ii), (vi) withm = 0; II. (i), (ii) , (iii). For these we shall write down the values of the stresses p~p, jTo, ^i, the components of the stress across the cylinder p — constant. In all, of course, we have zz = 0, and in I. in addition zp = zZ = 0. I. (i) f p = (^-V 1 +T- 3 z* P A cos w p ^ /a+1 - — - \ . 2 = ^ 2 p l +a-3z-p ■>) sin w The resultant is a force along Ox, of magnitude I I (^ cos w - p ^ sin u>)pda>dz , taken over the cylinder p , = ( a - H _ a+lV . . 2 h ■ 2y. = - 32*/* . I. (ii) 7? /a- 15 . — . _ A . 1 O, -^ — p 1 + a - 5 z-p 6 ] sm pai /a + J 75 „ _ ~~n~p +a-d Z-p -MCOSco The resultant is a force along Oy of magnitude — S'l-n-fxh. L(vi),«i«0. «-o I - = W = po> - Ip-p . THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 187 The resultant is a couple in the plane xy, of magnitude 8tt^/?, and we observe that the solution occurs with coefficient — Q^/StthJi. II. The stresses in the general flexural solution (94) are II. (i) -f- = (a - 3> V 2 F + P 2fl dp pa, dv 2p. dp • • f- = 2( z 2-7i2)|-v 2 F. 2/x dp j PP _ a ~ 7 z \ a P a+1 ? _/ a + 5 ,3_ 2fi 2 b 2h 4 " V 6 - 2/i% V s ' pa, = U ^.= 2(^-A»)p-i > 32 The resultant is a force along Oz, of magnitude - -~viih* . II. (ii) iS- = -I— ^ Zp - l COS o, - 2^— 7? - 2/^2 )p-» COS CO 2p 2 r V 6 > H pai 2p~ .+1 -2(" a "t 5 2 3_27 t 2 2 V 3 £ = 2(z 2 - 7i 2 )p- 2 cos co 2/j, The resultant is a couple about Oy, of magnitude I I {z(^ cos to - ^ sin (o) - p cos a^jp^cocfe, taken over the cylinder p, 32 3 II. (iii) 32 „ — 7rp/4 3 — - — 2 3 — 27A )p 3 sin to pp 7 — a , . ££■ = — - — zp i sin to 2p 2 ^ = ^lzp- 1 cosco + 2f a ±^2 3 - ! J£ = 2(* 2 - A 2 )p- 2 sin co . 2p The resultant is a couple about Ox , of magnitude I / { — 2 (pP s ^ n (0 + p<" cos w ) + P s i n w • p^}p f?co (7z = - — 7rp/< 3 (95) 31. Conditions for the existence of a solution with finite potential energy. Elastic equivalence of statically equipollent loads. The corresponding results for any distribution of body force, or of traction on the faces of the plate, may be deduced at once from the above by integration with respect to />!,«!, z 1 or p 1 , Wj with z 1 = dtzh. If the region within which the force is applied be entirely enclosed by a cylinder p = a, the results are valid for all points exterior to this cylinder. 188 Ml J JOHN DOUGALL ON AN ANALYTICAL THEORY OF For a distribution of force of finite intensity per unit area or per unit volume, the potential energy of that part of the solid within the cylinder is clearly finite. The energy of the remaining part of the solid can be determined from the forms of § 29. Now, the energy between the cylinders p = a, p=p is the integral of taken over the belt of the cylinder p = p cut off by the plate, diminished by the corresponding integral for p = a. Hence the condition of fmiteness of the whole potential energy is simply that the value of the integral for the surface p = p tends to zero as p tends to infinity. This condition is obviously satisfied by all the partial solutions of § 29, except those which have been already singled out as conveying a finite resultant stress. It is also satisfied by one of the latter class, namely, that which conveys a couple in the plane of the plate. Hence, when force is applied to a circumscribed portion of the solid, a solution giving finite potential energy will exist provided the force either constitutes an equilibrating system, or reduces to a couple in the plane of the plate. It does not follow, however, even for an equilibrating application of force, that a solution will exist giving vanishing displacements at infinity. We need only point to the solutions of § 29, II. (vi) and (iv), (v) with to ±= 2. This being so, it may be of interest to write down a few more details of those solutions which rank in importance next to the solutions of finite resultant stress. I. (iii) with to = 2. u-- { 4p _1 + (3 - a)z'7>- 3 } cos 2a> v = I - ^±V ] + (3 - a),r>- 3 } sin ! w = (3 - a)zp~- COS 2w pp_ ) I£= J (a - 7)p" 2 + (a - 3)3zV 4 - cos 2c 2u ( ) {^V 2 + (a-3)3 Z V- 4 | sil pa) _ 2/x This solution occurs with coefficient (X 1 x 1 -Y 1 y ] )/327riuh. I. (iv) with to = 2 is obtained by writing sin 2», — cos 2u> for cos 2w, sin 2w in the preceding, and the coefficient is (X 1 y 1 + Y 1 sc 1 )/32?r/*7j.. I. (v) with m = 0. U = p v = w = Z ' ^ I Coefficient = ■[ a -+ -(X^ + \\ ft) + (a - 3)z 1 Z 1 \ l32vfih. pa> — U I (2 j I II. (iv) with to = 2. F = | cos 2«. u — [ ( .' z 3 - 2/r.':;)p _3 cos 2w a+ 1 a + -££ = —^—zp - cos 2oj + a term in p 4 ■Zp. 2 o+l v= A zp~ x sin 2to + l-^-z* - 2hh V" 3 sin 2 W . ^ = - ^^zp' 1 sin 2 W + \ 6 / 2u 4 a + 1 , (a — 3 ., , w = - . cos _'o) + I z + a ) p - cos 2w ^- = 2(z'- - Jr)p 3 cos 2( THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 189 The coefficient is II. (v) with m = 2 is the above with sin 2«, — cos 2w for cos 2«, sin 2« and coefficient II. (vi). F = log^ 327^ 24 = - (a 4- l)zp ' V =0 !0 = (a+l)log^- f =(a+l>p- pw - p.: :-o. ) Coefficient is { -^X^Y^ + ^-^V^ For all the remaining solutions, the stresses are of the third or higher order in I//), The results of this and the preceding article bear directly upon a principle of fundamental importance in theories of approximation, generally referred to as the principle of the elastic equivalence of statically equipollent systems of load, and a study of these results will be found of service in imparting precision and definiteness to one's view of the principle in its application to the theory of plates. It may be noted here, with reference to the occurrence of the function log (p/2Ji) in some of the principal solutions of § 30, that it would make no essential difference if this function were replaced throughout by log (p/'c), c being any length whatever, the unit of length for example. The change would be equivalent to adding a solution of the permanent type, giving no body force or traction on the faces, and it will be observed that the addition would disappear altogether when the applied forces are in equilibrium. We have here, in fact, an instance of the indeterminateness that of necessitv arises in the absence of conditions at infinity, and we are thus brought to the question, what is the exact extent of this indeterminateness ? or, as it may be put, given one solution of a problem satisfying the conditions at a finite distance, what is the most general solution satisfying such conditions ? For the investigation of this question we have at hand a powerful instrument in Betti's Theorem, which occupies in the theory of elastic solids the place held by Green's Theorem in the Theory of the Potential. 32. Betti's reciprocal theorem. Verif cation of preceding solutions. Betti's Theorem may be thus stated : — Given two sets of displacements of an elastic solid, with the two corresponding sets of forces maintaining these displacements (including body forces, surface tractions, and kinetic reactions), then the work done by the forces of the first set acting on the displacements of the second set is 190 MR JOHN DOUG ALL ON AN ANALYTICAL THEORY OF equal to the work done by the forces of the second set acting over the displacements of the first. In potential theory one of the chief applications of Green's Theorem is to the case when one of the potential sj^stems includes a mass concentrated at a single point, and in the present subject Betti's Theorem finds an application of like importance when one of the displacement systems contains a finite force applied at one point, or, in analytical language, includes a point singularity of the first order, that is to say, of one of the three forms indicated in (6). Thus, let us suppose the solid to be bounded by a surface S, and in the first set let the displacements be u, v, w ; the components of body force per unit volume X , Y , Z ; and the components of the traction on S , F , G , H ; in the second set let the displace- ments be u r , v f , w' ; the only internal force a force X.',-Y', 71 at (x', y', z'), and the tractions on S , F, G', H'. We may apply Betti's Theorem to the space bounded by S and a sphere S' of radius e drawn round (x', y', z') as centre. Thus we have f f f(Xu + Yv' + Zw')dY + j j (Fu + Go' + Hw>iS + f [{Fu' + Gv + Kw')dS' = f f(F'u + G'v + Kw)dS + f f(F'u + G'v + H'w)dS' . Now take the limits of both members of this equality for e = 0. Since near the centre of the sphere S', u', v', w' are of order l/e, F', G', H' of order 1/e 2 , and cZS' of order e 2 , the effect on the volume integral is simply to extend it to the whole volume within S ; the surface integral J j (Fu' + Gv' + H.v/)dB' vanishes, and the surface integral I I (F'u + G'v + Ww)d& has the same limit as u(x, y, z')JJF'dS' + v(x, y', z) J JG'dS' + w(x', y', z')jfu.'dS', namely, u(x, y', z')X + v(x', y\ z')T + w(x, y', z')7J, the tractions F', G', H' on S' being statically equivalent to the force X', Y', 71 at its centre. It is thus apparent, and might indeed have been anticipated, that Betti's Theorem may legitimately be applied when one of the systems contains a force acting at a single point, provided the work done by this force on the other system of displacements be taken into account. The theorem thus becomes f f f(Xu + Yv + Zw')dY + f f(Fu' + Gv' + Kw')d$> - f f(F'u + G'v + K'iv)dS = u{x, y', z')X' + v(x, y, z')Y' + w(x, y, z')Z' . . . (96) In order to apply the theorem to the plate problems under discussion, take for the solid a portion of the plate bounded externally by any orthogonal cylinder. Let us THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 191 also suppose that the system u , v , w is maintained solely by tractions on the cylin- drical edge, and the system u', v' ', w' by such tractions along with the force at (x\ y r , z'). Further, it will be convenient to decompose the latter system, and take u x , v x , w 1 as due to a unit X force, u. 2 , v 2 , w 2 to a unit Y force, and u s , v 3 , iv s to a unit Z force. The corresponding tractions on the edge we will denote by X , Y , Z ; X x , Y x , Z x ; X 2 , Y. z , Z 2 ; X 3 , Y 3 , Z 3 . The theorem (96) then gives (97) u(x, y\ z) — I I (Xm x + Yv x + Zw 1 - XjM - Y x u - Zjw) JS N v(x', y ', 2) = I j(Xu 2 + Yv; 2 + Zw 2 - X 2 m - Y 2 v-Z 2 w)dS - . w(x, if, z) = I I (Xm 3 + Yv., + Zw a - X 3 u - Y. d v - Z 3 w)c£S the integrals being taken over the edge. As one application of these forms, we may indicate briefly how they can be used to verify the single force solutions already obtained. Take, for example, the case of a Z force, and let u s , v 3 , w s have the values defined in (63), (64), (65). Also let the edge be the cylinder R = constant. (i) The coefficient of the principal flexural term, in which, with the notation of (94) Foox(-R), is determined from the condition that the resultant of the stress zR must balance the applied force. It is interesting to note that the conditions of equilibrium of applied forces and surface tractions may be regarded as special cases of Betti's Theorem. We have only to take for auxiliary systems the rigid body displacements u = 0,v = 0,w=l; u = y , v= —x, iv = , etc. (ii) In the third of equations (97) take for u , v , iv the values of (94) with F = R 2 . Only the two flexural terms of (65) contribute to the surface integral ; the contribution from the particular solution <p = G kR sinh kz , 9= —cosh 2kJi'<P must vanish, as we see by pushing the edge to infinity. This, with the result of (i), gives the coefficient of the second flexural term of (65). (iii) The principal extensional term is verified by taking a+1,. _,, a+1, 9, . (x-x), v= (y-y'), w = (a-3)z, (iv) The coefficient of the particular solution <p = G kR sinh kz , 9= — cosh 2Kh'(p in (63) is verified by taking for u, v, iv the values defined by (p = J kR sinh kz , 9 = — cosh 2kJi '(p. None of the solutions corresponding to the other roots of sinh 2kJi — 2kJi contribute to the surface integral. In fact, the partial contribution from a root k' being inde- pendent of the radius of the cylinder, must vanish identically, since the Bessel Functions supply a factor tending to zero or infinity when R is made infinite, according as k' is a higher or lower root than k. (v) The coefficient of the particular solution <p — Q k~R cosh kz , 9 = cosh 2kJi'(J), may be verified in the same way. ;l l'.)2 MB JOHN DOUGALL ON AN ANALYTICAL THEORY OF It is now easy to see the significance of the forms of the coefficients in the solutions of § 29 and the confirmation of the values there given would obviously present no difficulties. 33. Finite plate under edge tractions. Form of the solution deduced by means of Betti's Theorem. We pass, however, to a more important application of the theorems (97). The system u , v , iv we still suppose maintained by edge tractions alone, but in addition to the external edge the solid may now be bounded by one or more internal edges. For "i » v i , Wi, etc., we take the definite values defined in (79), (82), (83), and in (63), (64), (65). Thus in (97) u x , v 1 , Wy , X x , Y x , Z x , and the other displacements and tractions marked with suffixes, are known functions of x', y', z', and the equations give explicitly the values of the displacements at any internal point in terms of the displacement and stress at the edge or edges. The ideal solution would give the internal displacement in terms of edge displace- ment alone, or of edge stress alone, but the analytical difficulties are such that we are unable to solve the problem thus completely even for the simplest case, that of a single infinite plane edge. Meantime, however, we may derive valuable information from the expressions of (97), and in the first place as to the form into which any solution due to edge tractions alone may be thrown. Just as in the case of the original source solutions, we find that the solution, in which, of course, the accented letters are now the variables, may be decomposed into an extensional and a flexural part, while in each of those parts we may separate a permanent mode from an infinite series of transitory or decaying modes of two types, the \J/- type, characterised by no dilatation or normal displacement, and the , (p type, in which there is no molecular rotation in the plane of the plate. In the following analysis integrals of the same form as those in (97) occur frequently ; the system u , v , w appearing in each case, but associated with various other systems. For conciseness we shall refer to the first integral of (97) as the work differ owe from Uy , i\ , Wy , and similarly in other cases. I. Extensional part of the solution. (i) Permanent mode. In Uy , v-y , Wy , the terms which relate to this mode are the unambiguous terms, even in z, of (83), after these have been divided by 47r,u(a + 1). These, as may be seen from a glance at the beginning of § 27, are equivalent to . _J_ d_f _J 2 . ^ ' Srr/Ji r/y'V* 2 " V X 1 / d ( 1,.,\ 3-a/l , 1 . 2 \ d ,, 32-n-fj.J/ V* standing for JU^ 2 THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 193 Now let the work difference from the system >A = g — Jx - -j z2 v 2 x) be denoted by E x , and that from the system = <f> = — — Jy - y*V 2 x ) by E 2 ; then obviously the work differences from the two systems immediately preceding are respectively ^i and —f + 3 "Y 1 e f ' - — ^A— V ' 2 E . <#?/' <ta' a+l\2 3 /eke' 2 ' Hence (97) gives In the same way from w 2 , ^ , w 2 and w 3 , v 3 , w 3 we obtain V(X V z) - - f ' E l + dK 2 + 3 - a f 1 /2_ 1 7, 2 \ <* ^'21? w(a y ,z) = -2 y Z E, a+ 1 Moreover, it can be seen in a moment that the displacements due to <A = -r-,v" 2 x and t0 = <£ = - — -^ 4-' v 2 x a# a-f 1 dx are in reality the same ; as also those due to It follows that i^ = — — ,v' 2 x and to 6 = d> = -^— — y' 2 y r dx A a+1 chj X dy a+1 dx and / /V '2 El --A_ 4v'% = dx a+1 dy ' J If we write U for and V for dE l (IE 2 _3-a Id , 2F dy' dx' a+1 J dx~' V 2 ' (iEj c?E, 3 - a 1, 2 d , 2 tj, efo' (/// a+1 3 oft/' 2 ' we obtain the form which it is convenient to take as the standard for this kind of strain, namely, .,/nt ./ j\ tt , 3-o 1 ,„d fdU dV\) u(x,y,z) = V + z 2 + a + 1 '1 dx \ax ay J «.m- v + .m^^4> <93) with a+1 Veto (/// / dy\dy dx) a + 1 dx'\dx' + dy) ~ d_(dU _ dY\ _ _8__ d_A7U d_V\ _ dx'\dy' dx) a+1 dy\dx' + dy')~ TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 30 194 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF (ii) Transitory modes, \f/- or rotational type. Referring to the expression for \J/ in (82), put E 3 = work difference from the system ^ = g — ^ cosn k^o" 5, • Then for this part of the solution u(x, y', z) = 2-i 2^-^ cosh kz K III/ v(x, y, z) = 2^(- 2)-=-? cosh kz w{x, ?/', z) = where a: is a pos. imag. root of sinh k]i, and -^ + -pf + * 2 E 3 = . The solutions here are obviously of the type ^ = cosh kz'~Ei 3 (x', y'). (iii) Transitory modes, 0-<p or dilatational type. Looking to (64), (82), put E 4 = work difference from the system [ j. = cosh kzG *R Then , , , ,. ydE A 8^(a+ l)»c 2 7t(cosh 2kA + 1) 6 — cosh 2kA •<£ 2kz' sinh kz' + (cosh 2kJi + a) cosh kz' <(x, y, z) = ^ kE 4 { 2k«' cosh kz' + (cosh 2k/i - a) sinh kz'} (99) (100) where k is a zero of sinh 2kJi + 2kJi with pos. imag. part, and -^ + -pf + k 2 E 4 = . The solutions are of the type (p = cosh kz'E 4 (o;', y'), = cosh 2k/V<£ . II. Flexural part of the solution. (i) Permanent mode. Let F x = work difference from the system 3 (■ -JL-8 2 32ir/Ji 3 V X 6 " V X I'-a^*-^-**^) Then V ,r $i = work difference from d>= — — — -, z V 2 x = - , and ,, , » , dF, 2 / tt + 5 a+11 _d „,_ '(to 1 / / / , N - ^F x 2 /a+5 , a+11 w(x, y', z) F, + { A/V - | 2 ' 2 + ~ (^ - ft')}V' 2 F 3 (101) THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 195 Here V /4 F X = , and if we write (« + 1)F for F : + ±h' V f *Fi , these expressions reduce to the form which we have taken throughout for this kind of strain, namely, / ' / -\ d u(x,y,z)= M v(x\y',z')=^r w{x\ y, z) = where V' 4 F = . -(a + l)(z'F-Az' 3 V' 2 F) + 2(^' 3 -/<Y)V'F (a + l)(F-|z' 2 V' 2 F) + 2(z' 2 - h 2 ) V ' 2 F (ii) Transitory modes, ^ or rotational type. Put F 2 = work difference from the system \^ = — ~ — sy sinh kzQqkR, . Then d¥_ 2 dy dy v(x\ y\ z) = 2- (- _ 2 ) ^J? si «"(*', 2/', z') = ^ ( - 2) — - > sinh kz' ,72 F ^/-F where k is a pos. imag. root of cosh kTi , and -v-rf + -j-?f + k2 F 2 = The solutions are of the type \^ = sinh kz'F^o/, y') . (iii) Transitory modes, 6-<p or dilatational type. sinh kzQ () kR Put F 3 = work difference from the system j 8 7 r / u.(a+ l) K 2 /i(cosh 2k/i - 1) I 6 = - cosh 2kIi • </> Then , , , , x y <?F 3 u(x,y,z) =Zd — 4- K dy 2kz cosh kz' + (a - cosh 2kJi) sinh k.^' (101') (102) (103) w{x\ y, z) = 2, kF 3 J 2kz' sinh kz' - (a + cosh 2/c7i.) cosh kz }_ where k is a zero of sinh 2*h — 2kJi with pos. imag. part, and -r-|-+ -r-jf + k 2 F 3 = . The solutions are of the type <t> = sinh kz' F 3 (x', t/) , = — cosh 2k/7 ■ . 34. Form of the solution for edge tractions deduced by another method. We have thus shown that the most general deformation of a finite plate under edge tractions only is compounded of the types specified in (98) . . . . (103). The deforma- tion is of the same form as that given by our infinite plate solutions for any part of the solid free from body force or surface traction, and it may be of advantage to show in a direct manner why this should be so. Suppose, then, that we have given a displacement (u, v, w) of a finite plate bounded by an external edge S and one or more internal edges S', the only applied forces being tractions on the edges. Imagine the plate continued inwards and outwards so as to 196 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF form a complete infinite plate. By the general existence theorem of the subject, there exist values of it, v, W in the space within an internal edge, continuous at the edge with the values of the displacements of the original solid, and produced by edge tractions alone. Similarly, if we take any surface S", within the infinite plate, but completely enclosing the edge S, there exist values of u, v, iv continuous with the original u, v, w at the external edge, and becoming zero on S" ; these also being produced by edge trac- tions only, namely, on S and S". If, then, we take u, v, w to be zero outside S", we obtain altogether a system of dis- placements continuous throughout the infinite solid. The forces required to maintain this system are given directly by the general equations of equilibrium. These forces form areal distributions on S, S', S", and are measured by the discontinuity of stress at these surfaces. Further, on the whole they make up an equilibrating system. But we have shown in the preceding pages how to find a solution for such a system of force, this solution giving displacements of order log R at most, and stresses of order Rr 2 at most, at a great distance. Only one solution fulfilling these conditions being possible, our solution is the solution. Hence, finally, any displacement of a finite plate under edge tractions only is of the same form as that given by our infinite solid solutions for a certain system of areal force, distributed partly over the edges, and partly over an arbitrary external surface. This is what we proposed to prove. 35. General solution for an infinite solid under any forces. It is now easy to determine the most general form of displacement of an infinite solid, under null body force and face traction, and free from singularity at a finite distance. For if u, v, w be any such displacement, then within any surface S, however distant, we have proved that u, v, iv are given by the absolutely convergent series (98) .... (103). If we take a right circular cylinder for the surface S, the functions F which satisfy equations of the form j- 2 + ,—, + k"F = can be expressed in series of the form ^ J m *p(A m cos Mini + B m sin wco) , TO and the only restriction on the coefficients A m , B m is that they must make the double series in which the complete solution is thus expressed absolutely convergent for all values of p, however great. The most general solution for any system of force applied at a finite distance is of course obtained by adding to this complete free solution the particular solution already investigated. It may be observed that this final result might have been obtained in one step by the process of § 33, if in that article we had taken for u, v, w any displace- ments under given body force and surface traction, instead of under edge traction only. The identity of the results of the two methods will be seen to depend essentially on the THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 197 fact that in the solution for a single force in any direction, the component displacement in that direction is symmetrical in the accented and unaccented coordinates, a theorem analogous to a well-known property of Green's function in Potential Theory. It is interesting to observe that, in the process suggested in the last sentence but one, we only need to know the comparatively simple source solution for a single Z force in order to deduce the w displacement for any system of forces and face tractions whatever. 36. Application of Betti's Theorem to the problem of given edge tractions. In the remaining pages, we shall be occupied almost exclusively with deformations of a finite plate under edge tractions only. For brevity we may refer to such deforma- tions as free. The formulse (97) express the internal displacements in terms of the edge dis- placements and edge tractions. We may indicate here the general lines along which we naturally proceed in the attempt to reduce these formulae to expressions in terms of displacements alone or of tractions alone. Taking the first equation of (97), for example, if we wish a formula containing edge displacements only, we look for free displacements in the form of functions u{, v/, w-[ of x, y , z, such that u x + u{, i\ + v/, ■w 1 + w{ shall be equal to zero at the edge. If X/, Y/, Z/ be the edge tractions in the system uj, i\ r , iv/, then by Betti's Theorem f ((Xtti + Yvi + Z< - X> - Y x 'v - Z/wW = 0, and by addition of this equation to (97), u(x\ y, z') = - ff { u(X x + X/) + v(Y 1 + Y/) + w{Z l + Z/) } dS. The problem of arbitrary edge displacements is thus reduced to a problem in which these displacements have a comparatively simple form. When we attempt to find a formula in terms of edge tractions only, the procedure is not quite so simple, in consequence of the fact that the tractions X t , Y l , Z x are not equili- brating, but equivalent to a negative unit X force through (x f , y', z'). From various methods of meeting this difficulty we select the following as the most convenient in the present case. We have seen in § 30 that the system i^ , v-^ , w 1 can be decomposed into four systems. The first system, say \J l , V x , W x conveys no resultant stress ; the second system conveys a stress equivalent to a unit X force through the origin, and the displacements are independent of x', y', z' ; the third system conveys a couple z' in the plane zOx , the displacements contain z' as a factor, but are otherwise independent of x', if, z' ; the fourth system conveys a couple — y' in the plane xOy , the displacements involving x', //, z' only in the form of the factor y' . 198 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF The displacements u. 2 , v 2 , w a and u 3 , v 3 , w 3 are similarly decomposable into equilibrat- ing systems U 2 , V 2 , W 2 and U 3 , V, , W 3 with other systems conveying resultant forces and couples. The contributions to u(x', if, z') . v(x', y f , z') , w(x', y', z') in (97) from the various systems conveying forces and couples amount on the whole merely to a rigid body displacement of the plate. If this be neglected, then the value of u(x', y', z') r for instance, becomes simply the work difference from the system \J 1 , Y x , ^SVi , the edo-e tractions due to which are equilibrating, and can be balanced by a free system U/, V/, W/. We then obtain from (97) u(x', y\ *')=(({ X(U X + U/) + Y(V, + V x ') + Z( W 1 + W/) } ^S and similarly for v, w. 37. Exact solutions of special problems for a circular plate. As already stated, we are not at present in a position to complete the solution of the problem of arbitrary edge tractions, even for the simplest form of edge. The method just indicated may be used, however, whatever be the form of the edge, to< obtain approximately the boundary conditions which define the permanent part of the solution. But before entering on this important application, we shall consider a few special problems which admit of exact solution. All of these have reference to a plate bounded by a right circular cylinder, with or without a concentric circular aperture, and to systems of displacement symmetrical about the axis. The radius of the external edge is a, of the internal edge b ; and the axis of z coincides with the axis of the cylinder, u, v, w are the displacements in the directions in which the coordinates p, «, z increase. Problem 1. Symmetrical transverse displacement. The displacement v, in the most general case, is given by a series involving cosines and sines of multiples of w. We can determine the symmetrical term of the series. This constitutes the whole solution when the plate is subjected only to symmetrical torsional force. For a transverse force Q x applied at the point (p x , ^ , z x ) we have seen in article 29 that the solution is . cU * = j a 1 dff U = ~ _ 7 1 d$ n 1 47T / U,(a+ 1) The solution for a constant linear distribution of transverse force on the circle p = Pi, z = z x , of intensity ^ 1 /27r/o 1 per unit length, is found by integrating this with respect to <t>i from to 2tt, and dividing by 2tt. The result of the integration is simply to THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 199 ■eliminate all but the symmetrical part of \^, and to eliminate , cp altogether. The solution is therefore, when p> p 1} _L £1 &TTpJl p + ~ — 7 2-i cosh kz y cosh kzJ '/c j o 1 G 'kp , (k a pos. imag. root of sinh kK) \ 0) - „ — ? ^j sinh K2 X sinh kzJ ' Kpfi^ Kp , (k a pos. imag. root of cosh k7i) when p <Pi, P and ^> x have simply to be interchanged. The only stress across a cylinder p is Hence if Q„ , 6 be the transverse components of the traction on p = a, p = b, and v x the transverse displacement at (p x , «j , z x ) produced by this traction, Betti's Theorem gives, as in (97), for the permanent part of v x , i \\ ^ = i ( ffnA + v^p W + -i- f fn b ±ds b . lir] o QirpJiJ J \ a a- J 8TTfjLhJ J p 1 Also, since the distributions i1 a , Q 6 have equal moments about Oz , affn a dS a + bjjih > dS b = . In the case of a symmetrical deformation, we have therefore ii- 1 ip.h\p l \(L-b\t Bl)1 n b dz+^ Va a 1 J J -h a The displacement i\ = p x being merely a rigid body rotation about Oz, the permanent solution is practically h 2 f h a 2 f h A ±- n b dz=--^—\ n n dz .... (ii> v,=- This might have been got at once by omitting the term v — p^ 8tt /mhp from the source solution, in accordance with the method explained at the end of the last article. For a uniform system of couple about the edge-normal JQ b dz vanishes, and the permanent displacement is rigorously null. For the transitory part of the solution we will, to simplify the algebra, suppose the cylinder solid. Further, this not being a case where the separation into odd and even parts in z is of much consequence, we may shorten the formulae by combining the two k series into one. Thus cosh k (h + z) cosh k (h + z x ) being obviously equal to cosh kz cosh kz-^ when 200 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF sinh kIi = 0, but equal to — sinh kz sinh kz x when cosh *h = 0, we may transform (i) if we put h + z = £ , & + Zy - {J , into v = - — — + ~ — r 2i C0S h K £i cos h K CJo' K Pi ( 3'o'' <: / ' ( K a P os - i ma g- root °f sinn 2k ' 1 ) • • (i") 07T/A/1 p 2trfjJl K For u = G Vp , we have when p = a, pw = - -^-(2G 'Ka + KaG Ka) . Hence, when the transitory part of v in (iii) is balanced, the solution becomes (p > p x ) v = - — - n + - — -Zu cosh *£, cosh «£J k Pi [G k P - " ° J Kp I . (iv> 07rp/i p 27rp/? k \ UJ k« + KaJ «a / This gives at p = a, J + - — - £j cos h «t,i cosh k^Jo'kpj t> = 8-jr/xh a 2-n-fjJi K V 2 J 'ko, + k« J n /<a Hence, for the free displacement at (p u z x ) under symmetrical transverse traction i2 a on p = a , Betti's Theorem gives (omitting the rigid body rotation) v x = -~2-i «T 'kp, cosh at J - — — )/ cosh xtCl a d'C . . . (v) 1 (Ji « ° ri _1 V 2J ' K a + KaJ Ka) J o s " s v 7 From this ( p( ,) 1= J_X ^ p + ,(pJ T oV cosh < f cosh *OV£ The series passes continuously, as p increases to a, into the limit — Zj cosh k£j / cosh K^f2 tt ^ , provided this latter series converges. By Fourier's Theorem we know that it does, namely, to the value & a , it being noted that I fi a d£ = 0. The solution is thus verified. Of course, it could easily be obtained by the Fourier method ab initio. The series (iv) converges very rapidly unless p and p 1 are nearly equal. By an application of the Residue Calculus, it may be transformed into a series in which the functions of p , pi are the fluctuating functions, and the functions of £ , £i the con- vergence factors. For consider the function of k, 1 cosh k(2/i - £) cosh k£ x , , / „ , 2G '»ca + KaG «a T , \ Vp.~ sinli 2k1i J "" ,, V lKr 2J 'Ka + KaJ oK a ° * P ) ' It is easy to see that log * disappears from the last factor, and that the whole function is a uniform, odd function of k; also that if ^>^i,jo>j°i, the function vanishes at infinity in such a way as to make the total sum of its residues equal to zero. The poles of the function are k — 0, the (pure imaginary) zeroes of sinh 2ich, and the (real) zeroes of 2J '/ca + koJ^cci. The function being odd, we have (series of residues at pos. imag. roots of sinh 2kJl) + (series of residues at pos. roots of 2J / /c« + K«J /ca) + |- residue at (k = 0), equal to zero. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 201 The first series of residues is the function v of (iv). Thus we obtain, (£> <^) 4 "V cosh K(2h - £) cosh k£j 1 T , T , ^ y ~^« sinh 2kK ^ a 3 ( J oKa) 2 J o«P 1 Jo^ I When ^>^,we have merely to interchange £ and (^ in this formula. We may verify in a moment from this, as from the perfectly equivalent form (iv), that the internal couple is balanced by the stress at the cylindrical boundary, and that there is no stress across the plane ends. But if we remove the last three terms from (vi), we make no change in the internal singularity, these terms being the same whether £ or £ x be the greater. We thus obtain the displacement when the internal couple is balanced at the plane ends, namely, (£>£i) 2 y , 4 "V cosh k(21i - 1) cosh kL 1 T , T , , ■•, TTfia* ^ s Tr/jLCt k sinh 2 K h K 3 a 3 (J Ka) 2 ° Hl ° ' v Here, as in (vi), the summation extends over the positive roots of 2 J '>ca + /caJ /ca . The solution for symmetric transverse traction & 2h , Q Q on the ends, which might be obtained in an abnormal form from (iv) with the cognate formula for p > p 1 , is given in normal form by a direct application of Betti's Theorem to (vii). Thus v (Pl . « - i- 2 cosh ffifa — i r w ,v P /xa K sinh 2kA K 3 a 3 (J /<«) 2 Jo , 8 y coahK(2h-L)J' KPl 1 /■» T , , + — ^ ■ , o' ; X-STt T2 V pJ K P d P \ ■ • ( vm > /xa * sinh 2k/j K J a 3 (J Ka) 2 Jo 4 /"" " 7—4 Pi^i "oP 2 ^P The result belongs rather to the theory of a long rod than to that of a thin plate. The permanent term depends only on the integral couple, and coincides with that given by Saint Venant's theory of Torsion.* 38. Problem 2. Boundary values of the normal displacement u, and the shearing stress normal to the plate ^>, are given functions symmetrical about the axis ; the displacement v, or the shearing stress ^, vanishes. We begin with the case of a solid cylinder. (i) Permanent extensional mode. Referring to § 33, I. (i), we see that under the conditions proposed the function Ei must vanish, and the solution in cylindrics is . ap 1 a + 1 ap l w (fti»»i) B3 ^Xi 2 iVi 2E 2 a+ 1 * The writer hopes to publish shortly a solution of the problem of equilibrium of an infinite circular cylinder, in which the celebrated solutions of Saint Venant will appear as the leading terms. It will be shown that in a fiuite cylinder the permanent modes are given exactly by Saint Venant's theory. In the theory of thin plates, the permanent modes can only, in general, be found approximately. TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 31 202 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF where E 2 = - - - . work diffce. from 8 = cj) = x(p) — |-z 2 log p + \px log p, or E 2 = — ^ — - . work diffce. from 6 = cp = log p , the part omitted being merely a constant. Now in the system 6 = <p = log p «-(o+l)/p ^=-2/t(a+l)/p» w = £=0 Hence the solution M= - (a+ l)p/a 2 p >2Ma-7)/a* w= -2(a-3)z/a 2 ^ = cfc . will, taken along with 6 = (p = log p , give it and p~z = at /> = a . The balanced solution gives p~p = — 1 6m/« 2 , w = 2(3 — a)2/a 2 , at the edge. Hence and w(p 1 ,z 1 ) = 2 — -gj 8a7i7 -a I 2/t J (ii) Transitory extensional modes. The solution is given by (100) with E ^ 8^ + lATcosh2K/ t + l) - W ° rk dlffCe - fr ° m the S ^ Stem {fr^S^ In the system mentioned M = /<G 'K|o{(cosh 2/<fe + a) cosh kz + 2kz sinh kz} ^ = K G Kp{(cosh 2i<h-a) sinh «2 + 2k2 cosh kz} __ IE = G 'Kp{(cosh 2kJi+ 1) sinh kz + 2kz cosh kz} — £^= - G Kp{(cosh 2k7i + 3) cosh kz + 2kz sinh kz} - — G 'K/o{(cosh '2kJi + a) cosh kz + 2kz sinh kz} • The balancing system for u and z^ at p = a is therefore tf>= — — ^ — J Kp cosh kz , 6 = cosh 2k/i<£ . Iii the balanced system, at the edge -, — < (cosh 2k1i - a)sinh kz + 2kz cosh kz > r kcl \ J .££ = — — ^ — < (cosh 2 kIi + 3)cosh kz + 2kz sinh kz > 2u aJ n kos I v J to- ot J I Hence for the free solution with edge values u = u a , T P = Z a , 9 ° (cosh 2/c/i - a sinh kz + T ^ash 2^ + 3 cosh* 2kz cosh kz) I , kz + 2kz sinh kz THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 203 (iii) Permanent flexural mode. The solution is given by (101) with 3 1 F, = 5 . — p, 2 • work diffce. from 6 = —<f> = z log p . 32-rrp.h 3 4 ^ = ^ = 2/x(o+l)z/p 2 0= -<£ = zlogp, W=-(a+l)z/p In the system w = ( a + l)log P Hence the solution balancing u and T P at /o = a is M=(a+l)z/o/a 2 to = - (a + l)p 2 /2a 2 + (a - 3)z 2 /ffl 2 « P = p P =2fjL(7 -a)z/a 2 In the balanced solution, at the edge to = (a + 1 )(log a - I) + (a - 3)z 2 /a 2 ; pp = 1 6p.z/a 2 The constant term in the value of it' will disappear since I zpdz = § . Thus F > - 3-4-= „•/*_{!<- V - ** }* and u (pi » z i) = - z iPi / \ 1 8,3-q 2 > ■ ■ a J ^(a-3)z 2 -8zw a V ^ a+l ' cfe (iv) Transitory Jlexural modes. The solution is given by (103) with F * = 8^(a + l)^(c OS h2^-l ) ' WOTk diffCe " fr ° m the SyStem { t= G °Zsh2 K h. 4> In this system u = kG 'i<p{ (a — cosh 2 Kh)smh kz + 2 kz cosh kz} w = kG kp{ - (a + cosh 2k1i) cosh kz+ 2kz sinh kz} ~2 k^- — (V*/°{(1 ~~ cosn 2kA) cosh kz + 2kz sinh kz] ~v o~~ ~ O Kp{(cosh 2kIi - 3) sinh kz — 2kz cosh kz} — - G '/cp{(a — cosh 2k1i) sinh kz + 2«z cosh kz) k P The system balancing u and zp at p = a is 4> = - -Yn — J Kp sinh kz , 6= - cosh 2kA • <p . In the balanced system, at the edge w — , , < - (a + cosh 2k1i) cosh kz + 2kz sinh kz > ao ku * ) ^ = — -^ — < (cosh 2/cA - 3) sinh kz - 2kz cosh kz > 2u ao n ko, [ ) sp. ao i Hence for the free solution with edge values u = u a , zp = Z a , J g^( - a + cosh 2kU cosh kz + 2kz sinh kz) \ _, -h \ + kuJS - cosh 2kJi sinh kz + 2kz cosh kz) ) If the given values of Z a , u a are the same as the edge values of z^ , u , in one of the particular solutions, then clearly this particular solution by itself is the solution, and the Y = I J 0*Pl C h { ^a 3 2(a+ l)/c 2 /i(cosh 2kK- 1) J 'ica ^ 2^ 204 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF integrals which define the coefficients of the other particular solutions must vanish, while the integral corresponding to the solution left has its value determined. These results are easily verified by actual integration. This remark may be used to find the solution for a hollow cylinder, which of course might also be obtained directly by the above process. We shall illustrate the method by finding the value of F 3 corresponding to any given root k of sinh 2k]i — 2kJi , when we have given u a , u h , Z a , Z 6 . This value of F 3 we know is of the form AJ Kp 1 + 'BG Q Kp l . The complete values of u and of ^/2m for p = p x are given by series which manifestlv converge uniformly so long as & < ^ < a . Multiply the series for zp~/2ju. by - (a + cosh 2k//) cosh kz + 2kz sinh kz , the series for u by k(3 - cosh 2*7t sinh kz + 2kz cosh kz) , add, and integrate with respect to z from — h to h . All the terms disappear except that associated with the given root k. We thus find (AJ ' KPl + BG '/cp 1 )2(a + l)/c 2 7i(cosh 2 K h - 1) — / )9~(p = p 1 )(-a + cosh 2kJi cosh kz + 2kz sinh kz) + ku(p = p x ) (3 - cosh 2kJi sinh kz + 2kz cosh kz) > dz . This is proved for the case b<p 1 <.a . Now take the limits of both sides for p 1 = a . The limit of the integral is found simply by replacing Tp and u (p = p^ in the integrand by zp and u (p = a) , provided the resulting integral has a meaning, which will be the case if zp and u (p = a) are integrable functions of z. Similarly we may take the limit for p x = b , and thus obtain two equations to determine A and B. It will be observed that by this method we avoid two difficulties which in problems of this kind are often introduced unnecessarily by physical writers, namely, (i) the difficulty as to the convergence of the series for 7 P and u, when the value p x = a or b is substituted term by term, and (ii) the allied difficulty as to the continuity of the series right up to p = a or b, even when it is known to converge. Judging from analogy, we may feel reasonably certain that the series will in fact converge at the limits, at least in the case of ordinary functions ; but it is worth while noting that, whether they converge or not, the Fourier method of assuming the continuity and convergence, and determining the coefficients by integration, does give the correct values of these coefficients. On the other hand, while our ' Green's function ' method proves definitely that any possible solution has the form given above, it does not prove that a solution is possible for arbitrary edge values of Tp and u. The investigation might be completed by verifying that the solution obtained does actually satisfy the conditions, which would not be difficult in the present case. Alternatively, we may rely upon physical considerations, or upon a general analytical existence theorem. The proofs of theorems of this type in other branches of physical mathematics have been considerably improved within recent years by Pom care and others, and their methods are equally applicable to the elastic equations. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 205 39. Problem 3. To determine the permanent modes, having given the symmetrical edge tractions z^> and p~p. Supposing the cylinder solid, we have only to make a slight modification in the process of (i), (iii) in last article. Extensional mode. In (i) the balancing solution must now be taken as _ ( a+1) 2 _p 2(3-a)(tt+l) £ U ~ 7-a a 2 ' W ~ ~ 7-a a 2- This, along with = cp = log p , gives p~p and p~z = at p = a . The balanced solution gives at the edge 8 a+1 2(3 -a) (a+1) « = - = , w— — s a I -a ' 7-a z Hence M (n, . Z. ^ = n. 1 i+l 1 M (pi » z l) = Pi j * ' " ' ' a+1 1 /•" . „ a - 3 } ~- fir-jr (4«P„ + a - 3 zZ a )<fe . (/°i. z i)= 2 ^n z ij ^-«i6m;-^ v This gives the permanent mode exactly. The ordinary approximate theory omits the term in Z a from the integral. In Chree's solution of the problem of a rotating disc, the stress zp vanishes identi- cally at the edge, while I F a dz = . His solution is therefore exact, so far as the fundamental mode is concerned. As in last article, we infer from the form of the above solution that for any other than the permanent mode / (i P ^ + a -3 z ? p ) = . This may be verified by actual integration. Farther, in the case of a hollow cylinder the solution is of the form m (pi>*i) = M + b /pi ) «-( Pl! Z 1 ) = 2A(a-3)z 1 /(a+l)} and the coefficients A , B are found from the conditions that I (4p • p P + a - 3 z • 7p) dz must, for p = a and p = b have the same value for the assumed form and for the given tractions. Flexural mode. Referring to (iii) of last article, the solution balancing 6 = cp — z log p at p = a is now „__(<*+l) 2 ZP (a+1) 2 P 2 (3-a)(a+l)z 2 W^ + 7-a a 2 ' 7-a 2a 2 " 1 " 7-a The balanced solution gives at the edge o a+ 1 z (3-a)(a+l) Z 2 u ~ _0 7 — ; w = const + = -» ( — a a I — a a 206 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF Hence «( Pl ,.:,)= -z lPl , The ordinary approximate theory takes account of the term in zP only. The solution for a hollow cylinder can be obtained by this method, or by taking, in the notation of (94), F = A P {~ + B log Pl + C (i Pl 2 log Pl - W) , determining C from the value of / TpAz at either edge, and then determining A and B from the conditions that I ( - 8 P z pp + 3 - a z'zp) dz must for P = a or P = b have the same value in the assumed form as in the actual displacement. 40. Expansions of arbitrary functions. When we attempt to apply the method of last article to the determination of the modes corresponding to the various roots of sinh 2/e/i ± 2ich, we are at once confronted with an apparently insuperable difficulty. The determination of any one mode is reduced by the application of Betti's Theorem to the special problem of balancing the particular source solution involving a given root k. Now in similar investigations connected with Laplace's equation, the equation of conduction of heat, and other partial differential equations of the second order which occur in physical mathematics, the analogous balancing problem can be solved without difficulty for certain simple forms of edge, and the balancing solution is of the same type as the particular source solution, that is, involves only the same root *. In the present problem, however, the balancing solution will in general involve particular solutions of all types, as will be seen below. Various theorems relating to the expansion of arbitrary functions may be found, similar to the theorem suggested at the end of § 38, but these do not help us, at all events immediately, to the general solution sought. One way of obtaining these expansion theorems may be indicated here ; the method is of very wide application. On a circular cylinder p — a, within the infinite plate, let areal force be distributed, the components of its intensity per unit area being P cos m« , Q sin m« , Z cos m<o , where P, Q, Z are functions of z. The infinite plate solution for this distribution of force can be written down, and the components of stress ^, ^ , ^ , calculated. These are given in different analytical forms according as p is greater or less than a. The expansion theorems are derived from the conditions of equilibrium with two similar equations for , Z. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 207 There are other expansions which we know must exist, but the coefficients of which we cannot determine. The following examples are of special importance, and will be found useful immediately. They refer to the case of a plate bounded by a single infinite edge, filling, say, the region a?>0, and the displacements considered are such that u, w are functions of x, z only, while v vanishes. Extensional modes. The permanent mode is of the form u = x, w= — ' z . In a transitory mode with (p = e iKX cosh kz,6 = cosh 2>ch ■ <f> , we have o — — 2 = ie lKX (cosh. 2k1i + 3 cosh k.c + 2kz sinh kz) o ,• 2 = e' Ka; (cosh 2kJi + 1 sinh kz + 2kz cosh kz) As a special case of the results of last article, it follows that the coefficient of the permanent mode is determined from the given value of I xx dz at the edge ; and this integral normal stress is zero for each of the transitory modes. Hence if P(z) be any even function of z, with J ~P(z)dz = 0, and Z(z) be any odd function of z, coefficients C K exist such that at the same time U 2 = q 4-1 C K j'e i,c:r (cosh 2/c/i + 3 cosh kz + 2kz sinh kz) = P(z) an U (i) x = o Zj C K e !,c:,: (cosh 2k?i + 1 sinh kz + Ikz cosh kz) = Z(z) Flexural modes. In the permanent mode F of equation (95) is of the form Aa^ + Ba: 3 . A and B are found from the edge values of f^z.Pxdz and T Zdz. These integrals vanish for a transitory mode, in which, with <f> = e iieX sinh kz,0= - cosh 2/ch ■ <p, XX 2/xiK xz 2~r~ K i = ie lKX (3 - cosh 2 K h sinh kz + 2kz cosh kz) — - 2 = e lKX (l - cosh 2k/i cosh kz + 2kz sinh kz). We infer that if P(z) be any odd function of z, with f zP(z)dz = 0, and Z(z) any even function of z, with j h _Z{z)dz = 0, values of C K exist such that simultaneously x = q^Li C K ie iKX (S - cosh 2k1i sinh kz + 2 K z cosh kz) = P(z) L^ = Zj C^^l - cosh 2k1i cosh kz + 2*z sinh kz) = Z(z) a; = (") The limit for x = may be taken term by term, provided the resulting converge. In the following analysis we shall assume that they do so, but this is merely in order to avoid lengthy forms of statement ; the argument could be put, if necessary, in a form independent of this assumption. 208 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 41. The problem of given edge tractions for a thin plate. The form of the complete solution is exactly known, and the three boundary conditions in their exact forms could, therefore, at once be written down. The whole strain is compounded of an infinite number of modes of equilibrium of known types, and it is obviously suggested as the method of attack that we should try to disentangle from the general boundary conditions those special conditions by which each mode is separately defined. When the plate is thin we find that within certain limits this can be done, and, in particular, the conditions defining the permanent modes, which in the case supposed are incomparably the most important, can be found with con- siderable exactness. We shall understand that the edge traction, or any component of the edge traction, is given as a function of x , y , z/h or of s , z/h , where s is the arc of the edge line, so that if £ be put for z/h the form of this function is completely independent of h. The theory may be applied to cases in which the proviso is not fulfilled, but before such application the given traction is to be separated into parts of ascending order in h , say, for example, f(x ,y ,'Q + h f(x ,y ,Q + h 2 f 2 (x , y , £) -f etc. ; then for a first approxi- mation we deal only with f(x , y , £). The theory does not contemplate such a distribution of traction, as, for example, sin (ms/h) , m being a number, where the rate at which the traction varies along the arc is of a lower order in h than the traction itself. The trace of the cylindrical edge on the middle plane of the plate is the edge line ; the outward normal, and the tangent, to the edge line will be referred to as the normal, and tangent simply ; the generator of the cylindrical edge at right angles to these at their point of intersection may be called the perpendicular. Let I, m be the direction cosines of the normal, then —m, I are those of the tangent. The normal displacement is p = lu + mv and the tangential displacement a = — mu + Iv . The tractions on the edge in the directions of normal, tangent, and perpendicular, are nn , ns , nz or N , b , Z. 42. Extensional strain. In this case N, S are even functions, and Z an odd function of z. It will be advantageous to express as far as possible the displacements and tractions at an edge in the various types of solution in terms of derivatives along the tangent and normal. Alongside the symbol a we shall use the more familiar <r, the relation between the two being given by « + 1 = 4(1 — o-) ; 3 — a = 4cr. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. (i) Permanent mode. U - {j+ \-cr2 dx\dx dil) , 7 , a- z" d / dU , dV 1-0-2 dy\ dx dij 'dU dV\ — -_ - M where 209 l-o- \ da; d ;/ / Put 2 ±(dU + dY\ + n_ v \±(dU_JV\ B0 ) dx\dx dy) dy\dy dx) \ 2—(— + — } -(l-o-) — (— -— ) = I dy\d.x dyj (ix\dy dx) A _ 1 /dU dV\ _ 1 fdu dv 1 - (T\dx dy / 1 - o- \dz dv/ n= j(*?_rfv \dy dx Then da; dy dA _ dil _ Q d?/ da; - and at an edge du dv dy dx dn ds dA_dII ds dn = Also 1 — cr Vda: dy -. 2u / du dv\ yy= \^VTx + ^y) ~ 2u l-o- /dw dv wy = ^ • — — I — -! l-o- 2 \dy dx/ The components of traction parallel to the axes are X = Ixx + lllxij ; Y = lxy + TWyy , These are easily transformed into Hence also or X = 2//zA +mn-- s j; Y= 2/*( - /H + m A + m = ZX + n»Y = 2uf A + m— - Z — V \ ds ds) r s = -mX + /Y = 2a( -U + l^+mtY ds ds) ,= 2/ /a- s.£.y \ as p ) s = 2 "(- n+ |-f)J TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 32 210 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF If P. Q be the values of p, q when z = , then -r, , <,dH /-w , 1 'yd A cis as 2yu. ds p 2 \ ds' p ds J 2fJL „ dP Q 1 2 /d 2 n ,liA = - IT + — - - -2. — —(TZ-\ -tr^- + ris £?S 2 rfs (ii) Rotational Transitory Modes. dy v=-2 ( '±\ dx 2/a cfa rfy 5. = ^-i. - ^t 2/x dy' 2 dx' 2/x, dy dz Taking the axes of x and y for a moment along the normal and tangent, these give at once by means of (k) n d\lS q=-2 d -± dn nn nd d\j/ ds dn 2dxjj p ds \ ns 27" dz 1 p # + 2 dn d->b nz . <*ty 2p.~ dsdz . The function \J/- can be expressed as a series of terms of the form ^„(a;, y) cos? ^ , where V fy» - ^r<A,< = ° • Hence in cases where the values of \^ along an edge are given independently of h, or generally, when the rate of variation of 4" along an edge is of the same order in h as ^ itself, say order zero, terms of various orders occur in the expressions for the displacements and tractions. Thus dj/ dhf, ds ' «V are of order dif/ d d\\i d d\j/ , dn ' ds dn ' ds dz d\f, dz 1 -2 It follows that in such cases this type of strain contributes mainly to the tangential displacement and traction at an edge. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 211 We see also that the principal part of the displacement is of one order higher in h than the principal part of the traction. (iii) Dilatational transitory modes. There would be some advantage in working with the functional symbols 6, (p, as with ^ in the last case, but on the whole it seems clearer to deal with a typical solution corresponding to a single root k of sinh 2<h + 2kJi. <£ = cosh kz/(x, y) ; = cosh 2k1i • <£ dx dy (cosh 2/c/i + a) cosh kz + 2kz sinh kz to = k/ { (cosh 1 Zk1i - a) sinh kz + 2kz cosh kz } ~= - K*f \ (cosh 2k1l + 3) cosh kz + 2kz sinh kz I — —hi \ (cosh 2kJi + a) cosh kz + 2kz sinh kz \ dy 2 I v J — - = — — - — \ (cosh 2k1i + a) cosh kz + 2kz sinh kz \ 2/x dx dy V J ~ = K^f- 1(1+ cosh 2k/;,) sinh kz + 2kz cosh kz \ 2u dx I J Hence nn = - k 2 / \ (cosh 2k/i + 3 ) cosh kz + 2kz sinh kz [ - ( f- + -~ ) \ (cosh 2kJi + a) cosh kz + 2kz sinh kz I \ p dn ds'J { J k~= \^r ^~~ — -r- ) 1 (cosh 2k7& + a) cosh kz + 2kz sinh i 2/x Vrts ara p $s / I <*// hKz\ 7T- = f-^ 1(1+ cosh 2«/i) sinh kz + 2kz cosh kz 2/x r/«. (. J Thus at an edge where the rate of variation of / is of the same order in h as / itself, say order zero, the normal and" "perpendicular displacements are of order -1, while the tangential displacement is of an order one higher ; the normal and perpendicular tractions are of order -2, the tangential traction being of order -1, or again one higher. Hence this type of strain contributes most to those components of displacement and traction to which the ^ type contributes least, at an edge. (iv) It is now possible to assign approximately to each of the three types of strain the portion which it carries of any given distribution of edge traction. Let this distribution be N, S, Z, functions of z, s, of order zero in h. We can satisfy the con- ditions to the first order by a solution in which the principal part of the traction 212 MK JOHN DOUG ALL ON AN ANALYTICAL THEORY OF is of order zero for each type of strain. For, taking account only of these principal parts, the equations to be satisfied at the edge are on this supposition N = N„ +N d , s = s p +s, z= z d the suffixes referring to the permanent, rotational, and dilatational types respectively. Now Also S r = 2^ and j ^ S r <fe = 2/z ~dz = 0. N d = - 2ix2-t k 2 / k < (cosh 2kJi + 3) cosh kz + 2kz sinh kz) > and, as in art. 40, Hence the above equations give j" A N„<fc = o. N„ = and these conditions determine the permanent mode, •v//- can now be found from the boundary condition For, taking the condition is 2Al ft = S-^f Sdz. az l 2hJ -n «A« (*, y) «os — , 2 fc, • tf cos ? |- Z = - ±j \ S - g^r/ Jl* } Now the right-hand member here is a function of s, z, even in z, the z-integral of which from — h to h is zero for all values of s. It can therefore be expanded by Fourier's Theorem in the form, valid from z = —h to z = h, > . n-KZ *— > A,, cos -7— »=1 ft ra'V 2 # ^ n is then determined as satisfying vfy* + -jr^n = throughout the plate, and taking the value A„/w' 2 at the boundary. Lastly, the equations to determine the dilatational mode are (since at the edge - - i K f to the first order), * ft ' K ~ j ( C0SU 2k ^ + 3 ) C0Sn KZ + 2kZ sin ^ KZ f = _ 9~ | ^ ' ~ 9T / -^^ f ./« ' **M (1 + cos h 2k/i) sinh *z + 2/cz cosh kz | = - h) Z THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 213 By the same method as in the case of \^, but using the theorem (i) of § 40, instead of Fourier's Theorem, we see that functions f K (x, y) exist, solutions of (y 2 + /c 2 )/*=0 and satisfying the above boundary equations. Thus the apportionment proposed for the edge tractions does actually satisfy the conditions to a first approximation. The solution found gives tractions of which the principal parts are the tractions actually assigned in the problem. The residual traction given by the solution is of the first and higher orders ; and a second approximation to the problem will be obtained by subtracting a solution giving the residual tractions of the first order, such solution being found by the method used in the first approximation. This process would be tedious, and the way would be blocked at an early stage by our ignorance of the coefficients of the expansion (i) of § 40. We therefore pass at once to the consideration of the powerful method furnished bv Betti's Theorem for the determination of the permanent mode. 43. Extensional strain. The Green's Function method for the permanent mode. The method has already been explained (§ 36). If we wish the permanent displace- ment at (x\ y', z'\ in any direction (say the displacement u), we take the permanent part of the solution for a unit force in that direction (a unit X force), modify it by removing the terms which convey resultant stress, and then try to balance it at the edge by adding a solution, without internal singularity, which shall neutralise its edge tractions. The displacements at the edge in the balanced solution, i.e. in the solution obtained as the sum of the source and balancing solutions, being u', v', w', or p', q', iv', and the given tractions X , Y, Z , or N , S , Z , we have c, y, z') = J f(Xu + Yv + Zw')ds dz the integral being taken over the cylindrical boundary. The thickness 2h being supposed infinitesimal, the object of the method is to deter- mine a few of the terms of p', q', w' of lowest order in h . An alternative method would be to determine the functions E 2 , B 2 of § 33, I. (i), in terms of edge tractions ; but the least confusing method of all is perhaps to determine v ,^dE, + r/E^ and y , = _<(K l + ,/K 2 dy' dx an " dx dx' These do not contain z', but when they are known the complete solution can obviously be written down. We begin with U', and in fact it will not be necessary to determine V separately. 214 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF Now U' is the work difference from the system From this system let those terms be removed which transmit the resultant stress (equivalent to a unit X force through x' ', y', z') . Then the remaining displacements have still the forms discussed in § 42 (i), and we shall use for them, and for the various quantities related to them, the notation given there, modified by the addition of a suffix in each case. The problem is now to balance the edge tractions due to the system u , v , w . The principal parts of these tractions, which in this case are simply the terms in- dependent of z , are balanced by a solution of the permanent type (which we shall distinguish by the suffix 1) such that at the edge (*» - 1°4") ♦ (*. - %-*) (-».♦$-*) + (-*+*-* <i> p J \ x ds p These conditions define the solution with suffix 1 . The residual tractions from the compound solution u + u t , v + v x , w + w x contain the factor z % and are of order h? as compared with those already balanced. To balance these residual tractions, solutions of all types are required, but, as in § 42 (iv), the per- manent solution (which will be marked with suffix 2) is determined from the integral residual tractions ; the permanent displacements are of order h, while those from the transitory solutions are of order h 2 , those from the source solution being of order hr 1 . The displacements of the balanced solution are therefore to terms of order h inclusive in the notation of § 42 (i), (2) and with these values of _//, q', w' U'(x',y',z) = jf(Np' + Sq+Zw')dsdz . . . (3) All the steps of the above process can actually be carried out in the case of a circular plate, and the final formula gives a perfectly definite solution provided merely that N , S , Z are functions integrable over the edge. It should be specially noted that, in this form of the solution, discontinuity of the applied traction gives rise to no difficulty whatever. On the other hand, the formula does not give a ready answer to such an important question as " What are the relations between the tractions actually applied, and the the error in each case being of order li? THE EQUILIBRIUM OE AN ISOTROPIC ELASTIC PLATE. 215 tractions required to maintain the permanent solution alone ? " or the practically equivalent question, "What conditions must the edge stress satisfy in order that the permanent mode may be absent from the resulting strain ? " The expression found for U' may be transformed so as 10 supply answers to these questions. In the first place, we note that the values of p', q', iv' will still be correct to the order stated (but will contain superfluous terms) if in the expressions just given for them we write n + Hj + n 2 for !!„ + II, and A + \ + A 2 for A () + Aj . Write also n for n + iii + n 2 , A ff -> r \ + \ + X, and similarly for the other quantities. Hence rv = — erzA and U'(x' , y' , z') = f f | N(P - W^) + S (Q + Joa^) - Z^A j ds dz . . (4) Also from (l) A - -^ - - , - n + — - " are of order h , and the formula for IT' will there- ' as p as p fore be correct to the same order as before if we substitute -^ + - for A , and ds p j--- for II. We shall also write as p f Nr7 2 = N n , f" 2 2 Ncfe = N .' -h J -h ]_ Sdz=S , f z 2 Sdz = S, f\zdz = Z 1 . Thus w>-j* {**-**&£ ->**+*«&%+*)-<% +!)} • w 7/N 2 , S 2 ,Z 1S ^rr, -^ 2 are continuous over each edge line, integration by parts transforms this into Hp(v^-i.«.-^f)^(M.f-rlH!.5)} . . (6) 216 ME JOHN DOUGALL ON AN ANALYTICAL THEORY OF ITcnce, in order that the permanent mode should be absent from the strain due to N, 8, Z the following two conditions are sufficient : — W - * Z -hr —* -A fr - S ' 2 - ft as- f> as o '/Zj . <r ^N„ . rf-'S, b + tr - l i -^ + Ao- — -v 1 = U as o as as - (7) at every point of the edge line or lines. Further, all systems of traction for which the left-hand members of (7) have given values at every point of an edge will produce the same permanent mode. Now as one such system of traction we may take the traction due to, or producing, the permanent mode by itself, as given in § 42 (i). This gives at once the boundary conditions satisfied by the functions U, V of that section, and these boundary conditions, with the internal equations dA cm _ Q . dA _ <m = dx dy ' rt.c dy completely define U, V which are thus determined, to a third approximation in general. The defining differential equations and surface conditions being practically of the same form as in the familiar first approximation, we need not detail the proof that U, V are actually determinate from the conditions, but pass at once to the important conclusion, an immediate consequence of this determinateness, that the permanent strain will not be absent unless (7) are satisfied, or, in other words, that these conditions are necessary ; as well as sufficient. From this again it follows that these conditions are fulfilled, to the order stated, by each of the transitory modes ; and this remark is valuable, because, once it has been verified by direct integration, it obviously leads, by an extension of the process of § 42 (iv), to a completely independent method of dealing with the whole problem. The method is noticeable for its simplicity and directness, but a somewhat serious defect is the difficulty of adapting it to the case when the edge stress is discontinuous. This leads us to consider the correction that must be applied to the integral (6) when the conditions of continuity stated in connection with it are not fulfilled. It will be sufficient to take a case in which breach of continuity occurs at only one point E of the edge line. We have defined the positive direction of an edge line in (k) ; let the excess of the value of f(s) just on the negative side of E over its value just on the positive side be denoted by [/(•?)]• Then if (p(s) be continuous J/(S) I $( S )dS = *(*)[/(8)] - j #0 £/(8)d8 , the integrals, we may suppose, being taken round the edge line from the positive to the negative side of E, and the value of <p(s) in the integrated term being taken as at E. THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 217 Then to (6) we have in general to add -Hf-7) [NJ+ K§ + 7) [S * ] -' Q[ZJ 1 or, otherwise arranged, p { if p« +, 4^]} + li {- * [ZJ + J 7 [N J " K * 1 } -fWJ + SMW The various terms of this expression may be interpreted with the help of the concep- tions of sources and doublets. Thus, to go back to (3), we see that the part of U' arising from an element N e&? of normal traction at E has (P + P^NoC&j for principal term. P + P x is therefore (principal term of the) value of U' due to a unit element of normal force at E. (Since this unit element can only exist in any actual deformation as part of an equilibrating system, the phrase due to in the last sentence must be taken under reservation. The solution of which Po + Pj^ is the x-displacement at (x f , y', 0) is in fact maintained by a unit element of normal traction at E, acting along with a continuous system of force in equilibrium with this element, and distributed over the edge in a manner depending only on the statical value of the element, and not at all on the position of E. For any equilibrating combination of elements, the aggregate of these continuous systems will disappear.) Now the first of the above integrated terms is equivalent to (p +p )£— -[S.,]. Hence the discontinuity in S 2 at E has the same effect at a distance from the edge as would have an element of normal traction distributed over the perpendicular at E so as to give a resultant h— [S 2 ]- Again an element — A of normal traction at E, combined with an element A at E',. where EE': ds, will give U' = ■A-fP. + PJ* ds (P + P x ), if we take Ads = 1 . — ( p o + p i) i s therefore due to a unit doublet of normal force at E, and from the term us |o-[N 2 ] we conclude that the discontinuity in N 2 at E has the same interior effect as- a doublet of normal force at E of strength — ^o-[N.J. The other terms may be interpreted similarly. It does not seem possible to account on physical grounds for any except the principal terms of the solution given above. The principal terms are of course the same as those deduced in the ordinary theory from the ' Principle of the elastic equivalence of statically equipollent systems of load.' TRANS. ROY. SOC. EUIN., VOL. XLI. PART I. (NO. 8) 33 218 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF With reference to the equivalence of mere discontinuities to line elements and doublets one or two remarks may be made. Discontinuity in the applied force will not produce infinite displacement at a line where it takes place, but a line element of load, and, a fortiori, a doublet, will do so. The permanent mode may therefore contain infinities at the edge which do not exist in the exact solution. There is really no difficulty in this, since the permanent mode does not purport to represent the strain, even approximately, in the immediate vicinity of the edge. The point may be illustrated by the permanent part of the infinite solid solution for a single force. This becomes infinite on the perpendicular through the source in a totally different way from the exact solution. A good deal of discussion took place at one time over a similar point in the fiexural solution. This will be referred to again, but the considerations we Jiave adduced seem to remove the chief part of the difficulty. 44. Fiexural strain. In this case N, S are odd functions, and Z an even function of z. (i) Permanent mode. This mode is defined in terms of one function F of (x , y) satisfying V 4 F = , and may be referred to simply as an F strain. tf,' = - (zY - ^ 3 v 2 F) ; 6 = zY - £z 3 v 2 F - 2hh^Y dn ? = to = ds 4(1 - <r)(zY - £z 3 v 2 F) + 2(l;: 3 - /i%)v 2 F 4(1 - (r)(F - kVF) + 2(?: 2 - /i 2 )v-l' For shortness in writing out the stresses, we shall work with symbols ^ , &, , -9 3 denoting operations of differentiation applied to F, and defined by the equations Then , 1 = 4v 2 + 4(.-l)(lii + | 2 S \= 4(1 - cr)( j- ~ - 1 -f \as dn p ds S., = 4 -f v 2 dn ii.n /x2 2 v ' s 1 J_ 2"1- THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 2.19 (ii) Rotational transitory modes. These are as in § 42 (ii), except that now where V , / x • 2n + 1 vz n=0 z n (iii) Dilatational transitory modes. <f> = sinh kz -f(x ,y); = - cosh 2k/i • <£ As in § 42 (iii), (a - cosh 2k/i) sinh kz + 2kz cosh kz I df * ds w = k/ { - (a -t- cosh 2kJi) cosh kz + 2kz sinh kz} 25 = - «y{ (3- cosh 2k1i) sinh KZ + 2K2 cosh kz} — -^- -\ \ ) < (a - cosh 2k1i) sinli kz + 2kz cosh kz > p ^ra as- / ! J = ( — — — _£ ) J ( a — cosh 2kJi) sinh kz + 2kz cosh kz > \ds dn p ds/ { ) , (1 - cosh 2k1i) cosh kz + 2kz sinh kz } 2fj. dn\ x ' f nz _ df We have -f- = - i K f to the lowest order, dn or /= i df k dn Hence if we put the above strain gives df dn y * i{(cosh 2k/i - 3) sinh kz — 2kz cosh kz) = N(kz) (1 - cosh 2k7i) cosh kz + 2kz sinh kz = Z(kz) 2/* I with an error of relative order h , m = ) ™ = g K Z(Kz) exactly, and we may note that I zN(kz) = and I Z(*z)dz = 0. (§41.) The same remarks as in the extensional case might be made here about the complementary character of the types (ii), (iii) in regard to their contributions to edge displacement or traction, when h is small. (iv) If we follow the lines of the discussion of the extensional case, we have now to consider the approximate allocation of a given system of edge traction among the three types of strain. 220 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OP The investigation is this time more complex, chiefly in consequence of the presence of the stress nz in the permanent mode. Since, moreover, flexure is much more important physically than extension, we shall give a fairly detailed discussion in the next article, but in the meantime we may examine what could be done with a solution in which, as in § 42 (iv), the principal part of the edge traction is of order zero for each type of strain, and the parts of higher order are neglected. Such a solution would give N = N, +N (j \ s = s,, + s, z = z lt j But we see at once that we do not in this way get a perfectly general distribution of N , S , Z , since the last equation gives / h Zdz = . A closer examination is therefore necessary, and it will perhaps conduce to lucidity if we consider separately the three cases of normal, tangential, and perpendicular traction. 45. Flexural forces. (i) Normal traction. JO I N being of order zero, we can satisfy the conditions by taking zF, -p , and g K all of this order, but besides the terms of order zero in the stresses, it will be necessary to take account of the terms of m which come from F and -v//-, albeit these are of an order one higher. Then -o .;- =-,^f + % k n(kz) . . . (i) =-** 2 F + § . (2) Assuming these provisionally, multiply (l) by z and integrate from —h to h. *>*-&'/->* ■ - ■ « From (2), since ^ is odd in z, and -^ = for z = ± h , we get ^ = a;. 3 -|/r^.,F . . . (6) In (3) the terms are of different orders ; thus, with the help of (6), £ 3 F + J^F = . . . (7) 1 9k Z(kz) = . ... (8) (4) and (7) define F, (6) then gives the edge value of ^ , and (5), (8) determine c/ K , the functions to be expanded obviously satisfying the conditions of § 40 (ii). THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 221 Further, with the values of F , ^ , g K so determined, the equations (1), (2), (3) are all satisfied, the traction nz vanishes exactly, and the residual tractions nn ns are of order 1. (ii) Tangential traction. 8 being of order 0, zF and -rf will again be of this order, but it will be seen that g K is of order 1. For, making these suppositions, we have = -.^F . . . . (1) #=-^F +& .... (2) 2/x - dz = £( Z *_7* 2 )£ 3 F + ^ + % Uk Z{kz) . . . (3) From(l), ^F = . . . (4) From (2), ^ f S&= - W ~ &P F + ^ • • • (5) Differentiate this with respect to .9, and subtract from (3). Thus " ^/_ a S ^=^ 2 -^)(^ + ^2F) + S^Z(k Z ) .. (6) and, integrating the last from —h to h, (4) and (7) define F, and (5) integrated from to z gives \J/. With the values so determined, and with (6) satisfied by g K (and this is possible in virtue of (7)), the equa- tions (l), (2), (3) are all satisfied, the traction nz vanishes exactly, and the residual tractions nn, ns are of order 1. By combining this with the preceding case, we see that the results do in fact give a first approximation to the solution, since the residual stresses N , S are each of an order higher than their original given values. The additional equation required to define g K might be found by carrying out the process of (i) with the residual normal traction 4/xyy . The analysis would be prac- tically the same as will be given in connection with the next case, (iii) Perpendicular traction. Z being of order zero, we shall have to take zF and -^p of order — 1 ,' g K of order 0. On this hypothesis, we shall write down the exact expression for nz , and the terms of order — 1 and in 7 m and ^-. We are then to have = -2S X F = -^ + § (I dxl/ + 2 ^^ + ^N(k 2 ) . (1) 2 dip /n'\ p dn ' ^' ^ =£(*>- 7, W + ^+ 2 !/k Z(kz) . . (3) 222 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF In the first two stresses the terms of order — 1 are annulled by taking ^F = (4) ^-(^-JMO^F ... . (5) Then integration of (3) from — h to h gives ^ + ^ 2 )F=-4^/;> .... (6) and from this with (3), Another equation is required before g K is defined, but in view of the results of (i), (ii) we can already infer that the values of F , ^ determined by (4), (6) and (5) are the correct principal values, since the residual stresses nn , ns are of order zero. As for g K , we note that we cannot complete its definition by annulling the residual stress nn , for the condition J _ h 2 -=- -^ zdz = is not satisfied. In order to get the remaining equation for g K we must therefore solve by the method of (i) for the residual normal traction 4/iT t- , so as to get the equation corresponding to (i) (5). The matter might be left at this stage, but it may be interesting actually to carry out the process of balancing the residual parts of nn and ns. By so doing we shall not only define g K , but also obtain a second approximation to F and \^ . We have to introduce into the solution F', >//, gj with z¥' and -X of order , gj of order 1. The equations to be satisfied are d dh \ - 2 dTJn- *-*(">- -V" • • • W _2 (fy r/y p dn - j -r ^ 2 . \ ; =^_/^,F'+ ^ + S<7«'Z(k Z ) . . (10) We must first find the principal value of -y in terms of defined quantities. Now from (5), if/ = (£z 3 - \hh)$.$, at the edge. Hence, by Fourier's Theorem, 32 ,,n tV • « 1 . 3tTZ 1 . 5tt2 \ . *= - V* *■*** [ Sm 2h ~¥ sm Wi+fr sm 2h -■'•••) Within the plate, therefore, 32/ . « 1 . 3-ttz 1 . 5tt2 \ THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 223 where and y^ m has the edge value £ 2 F. Thus ,, (2m + l)V, A — = - — r ft -I- 2 - sm — - - — A -P sin — — + . . . dn it" \ dn 2ft 3 4 dn 2ft the principal value of which is Let then and I67911 -r?/ ■ TTZ 1 . 3tTZ , 1 . 5ttZ — //,-j,i sin — - —sin -+ — sin — 7T 3 2 I 2ft 3 3 2ft 5 3 2ft ai n.\ 16/ • 7T2 1 . 3ttZ , 1 57T2 ^/ft)._ sin_--sin_ + _ S i 1 i- lr/3(z/h).%F J -a arc 7r" \ 3° 5 B / if = - 1 f 75 ft^,F 7T 7." »+-£+£+ Now multiply (8) by 2, and integrate from —h to A. Hence ^F'=-^y 5 ft^^F 7T S OS and 2<7 k N(kz) = J 2Vp(z/h) - 384-^V f -* <>,,F (11) (12) (13) (14) (15) This, and equation (7), define g K . As we do not require gj, we will eliminate it from (10) at once by integrating. Thus '-"W Multiply (9) by 2 and integrate. Then -If z # P i -A ,^ 2 =-^ F f and from this with the last or, from (13), ds\ p J -n dn f 3 V 2 ds ' ^F' + ^ 2 F'=-^iy 6 ft-|/- ] (16) (14) and (16) give F, and ^' may be found from (9). 224 MH JOHN DOUGALL ON AN ANALYTICAL THEORY OF If we write F for F+ F', we get from (4), (14) ds neglecting the term in JiF', which is of order h 2 relative to F. This may be written (*+3S*a*>-° • • < 17 > Similarly from (6), (16) we obtain /„ . d a , 384 . d 1 a \— 3 /"" r/ . We may regard (17) and (18) as the equations giving F to a second approximation. If we combine the results of the three cases of this article, we obtain 3 <•" 3 3 F + — &,F= - JL_J f Zr7z + — f" zStfe I 'As" 4jU.A 3 ( 7 -7l ffS .' -7l J (19) These are the equations usually referred to as KirchhofTs boundary conditions. The extension of the more approximate conditions (17), (18) to the general case will be given in the next article. 46. Flexural .strain wider given edge tractions. The Green's function method for the permanent mode. The displacement at (x f , y', z') due to tractions N , S , Z is defined in § 33, II. (i), in terms of the work difference from the system <£= - (3/32^) («x-Wx) e = (3/32^^) (2x _ i 2 3 V 2 x _ . 2] - Wx) From this system let the terms conveying resultant stress be removed ; the residue is still an F strain with F = F say, and F is of order h~ s . We have to balance F at the edge, and the edge displacements in the balanced solution being //, q', id', the work difference required (F 1 of § 33) is I I (Np + Sq + Zw')ds dz . The problem is to determine p', (/, tv' as closely as is practicable. The tractions to be balanced are ' r ' u Lwitli terms ui2 d 'Aj-1,,.= |(.r-/r)^F . THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 225 With tractions of these orders, of quite general form in z, the analysis of last article would lead us to expect that in the balancing solution F would be of order — 3, ^ of order 0, and g K of order - 2. But in consequence of N being simply proportional to 2, it will be noticed that § 45 (i) (5) gives 2(/ K N(/c2) = 0, and it follows that in the present case g K will be of order — 1 at lowest. The displacements p f , w' derived from the strain defined by the functions g K will therefore be of order 0, and q' of order 1. Thus we may anticipate that the first two terms of p', and the first three terms of q' and w' will be obtained in practicable forms, i.e. independently of series associated with the zeroes of sinh 2k.1i — 2xh. The tractions written down above may be balanced approximately by strains F', \f/. We require 1(*»-W 8 (F + F) + *V' dsdz These are equivalent to ^(F + F') = (^ + iU 2 )(F + F') = (1) which determine F', \^'. The principal terms of the residual tractions are j ns _ 2 d{j/' 2fx p dit (2) /3(z/h)& 2 (F + ¥') as in (iii) (12) of last article, and they may be dealt with in the manner illustrated there. 72 / " To balance them take strains defined by F", \f/', g K " with zF", -Vy , g K " of order - 1. We must therefore have 2h^(z/h)^ 2 (F + F) = - z^F" (2h*l P )ftz/h)» a (E + F) = - e& % If* + % IXZ o = hz2-h*)$ 3 r+f£ 2 ds dz From these, as in last article, 384 d TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). + ?</ k "N(kz) + 2g K "Z(«z) ttz 1 . 3ttZ (3) (4) 34 226 MR JOHN DOUXIALL ON AN ANALYTICAL THEORY OF whence '!£=- Vf3(z/h)W" + - ( I z 8 -* \hh ] .■'.( I (5) We are now left with tractions «b , ns of order zero. In the solution balancing these, ¥'" will be of order — 1 , >//" of order 2 , g"' of order . We do not think it worth while to write down the equations defining F w , but it should be noticed that they can be found explicitly. In fact, although the residual tractions nn , ns with which we are now dealing are partly defined by k series, the integrals I znndz and / Znsdz will be found to vanish to the order concerned so far as they come from these series, in virtue of the relation ^gJ"^{ K z) = , which follows from (iii) (7) of last article. The functions g K " give terms of order zero in p', w' '. p' terms of orders —2,-1 q' -2,-1,0 Hence, including in w we have p'=-4(l-^(F + F + F") q = - 4(1 - cr)a|.(F + F' + F" + F") + 3,-2,-1 4-2<r r ( 6 ) w = 4(1 - <r)(F + F' + F" + F") + 2(crr - /r)y 2 (F + F') . The value of -J- to a second approximation is gl = - h^(z/h)S 2 (F + F) - 1(1 Z 3 _ 1/^^ 2 (F + F) (7) and 4n is given by (5). The function Y x which (§33) defines the permanent solution is // (Np' + Sq + Zw')ds dz (8) For the case of a solid or hollow circular plate all the quantities in the right-hand members of (6) can actually be calculated, and we thus obtain the solution for normal traction to a second approximation, and for tangential or perpendicular traction to a third approximation, in a form, moreover, applicable without modification or addition even when the given edge stresses are discontinuous. We conclude by deducing the equations corresponding to Kirchhoff's boundary conditions to a second approximation. (They might be found to one order higher in the case of vanishing normal traction.) We suppose that the given tractions N , S are of the same order in h, and that Z is of an order one higher. Any case may be reduced to combinations of cases satisfying this condition. THE EQTJILIBKIUM OF AN ISOTROPIC ELASTIC PTATE. 227 4(l-o-) ds 0) If we write F for Fq + F' + F", then it follows from (6), (7), (8) that the terms of two lowest orders in Fj. are given by dnj -h ds) -h J -h + (d**_Lf\r WfK ,/ h)6d . \ds an p as I J -h As in the extensional case, § 43, the integral with respect to s may be modified by cZF means of integration by parts so that only F and ^ appear under the integral sign. Write /* zNdz=N 1} j h zSdz=B 1 , f Zdz = Z J -h J -h J -h f l 2h 2 (3(z/h)8dz=S 2 . J -h Then if N t , S x , Z , S 2 are continuous functions of s , Hence any two systems of traction will give the same permanent mode to .a second approximation, provided the values of ^i + -jj and ~^i +Zu+ ^ JL are the same for the two systems. Now for the system F x , § 44 (i), (10) ds ds ds Nj = - -i M 3 ^iFi , Si = - I-/AV.F, , Z = - £ M^i , S, = - I_^ 3 ^ 75 M 2 F, , § 45 , (iii) (13). 3 7T (12) Thus, with #! , $., , $, defined as in § 44 (i), the boundary conditions are -**(w + £»»£v,)-*.+t r/.s ds p ds ds p j (13) When only the principal terms are retained, these reduce to Kirchhoff's conditions. If S x or S 2 is discontinuous at any point of the edge, integrated terms will appear in equation (11), as in the extensional case, § 43. Thus, if the normal couple S x be discontinuous at a point P (s = s'), there will appear on the right of (11) a term F(s') | S : s = s + The method of dealing with such a discontinuity in any actual problem is obvious, for by (9) its effect is the same as that of an element | S x [ of shearing traction applied at P, a result which on the ' elastic equivalence ' theory may easily be obtained by a trifling modification of the process by which Thomson and Tait reconciled the con- ditions of Kirchhoff and Poisson. 228 THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. Addition to Paper by J. Dougall on — " An Analytical Theory of the Equilibrium of an Isotropic Elastic Plate." (Note added May 20, 1904.) The kindness of one of the referees enables me to supply the following references to recent work bearing on the subject of the paper. (a) J. H. Michell, in a paper " On the direct determination of Stress in an Elastic Solid, with application to the Theory of Plates," Proc. Lond. Math. Soc, vol. 31^ 1899, shows how the stresses might be found without previous determination of the displacements. In the case of the stress zz or R, he finds that y 4 R is a given function in the body of the plate, while R and d^Kjdz are given on the faces. If we neglect the conditions at the edge, which have practically no influence on the result, a value of R satisfying these conditions can be found, in terms of Fourier integrals for instance. Mr Michell does not determine R — this has been done in the present paper — but proceeds to deduce the forms of the remaining stresses, and the differential equation for the normal displacement of a point on the mid plane. One special case of normal force is worked out to a first approximation, and Lagrange's equation for this case deduced. For the conditions at the edge, reference is made to the ordinary Thomson- Boussinesq theory, which uses the principle of equipollent loads. (b) L. N. G. Filon, " On an approximate solution for the bending of a Beam of rectangular cross section under any system of Load, with special reference to points of con- centrated or discontinuous Loading," Phil. Trans. R. Soc. Lond. (Sec. A), vol. 201 (1903). Dr Filon's solution applies to a beam in which the ratios of breadth to depth, and of depth to length, are both small. The axis of z being taken in the direction of the breadth, the stress zz is taken as negligible, and equations are deduced for the mean values, across the breadth, of the displacements u, v. These equations are the same as equations (90), page 182 of the present paper, with the body force null. In order to see the reason of this from our standpoint, we may notice that the assumption that 22 vanishes eliminates all the solutions of what we have called the dilatational transitory type, and that taking the mean of the displacements eliminates all the flexural solutions, as well as the rotational transitory solutions. As regards the conditions at the ends, the beam is treated as a long rod. It may be of interest to remark that the results of § 43 above furnish the data for a more approximate treatment of the problem on the lines followed by Dr Filon. (c) A note appended to a paper by Professor Lamb in Proc. Lond. Math. Soc, vol. 34 (1902), pp. 283, 284, contains a solution of a special case of the problem of face traction. (d) In connection with existence theorems relating to the elastic equations, reference should be made to the work of Italian elasticians, as Somigliana, Lauricella, and Tedone - PRESENTED 9 FZB. jyui. The Transactions of the Royal (Society of Edinburgh will in future be Sold at the following reduced Prices : — Vol. VI. l'riee to the Price to i Vol. Price to the Price to Public. Fellows. £0 9 6 Public. Fellows. £0 11 6 XXXIX. Part 1. £1 10 £1 3 '< VII. 18 15 Part 2. 19 14 6 VIII. 17 14 Part 3. 2 3 1 11 IX. 1 17 Part 4. 9 7 X. 19 16 XL. 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December 1902. — Volumes or parts of volumes not mentioned in the above list are not for the present on sale to the public. Fellows or others who may specially desire to obtain them must apply direct to the Society. As the Society reprints from time to time parts of its publications which have become scarce, the absolute correctness of this list cannot be guaranteed beyond this date. TRANSACTIONS OF THE ROYAL SOCIETY OF EDINBURGH. VOLUME XLI. PART II.— FOR THE SESSION 1904-5. CONTENTS. Page IX. On the Measurement of Stress by Thermal Methods, with an Account of some Experiments on the Influence of Stress on the Thermal Expansion of Metals. By E. G. Cokbr, M.A. (Cantab.), D.Sc. (Edin.), F.E.S.E. ; Assistant Professor of Civil Engineering, M'Gill University, Montreal. (With Two Plates), . . . . . .229 (Issued separately 2nd September 1904.) X. On the Spectrum of Nova Persei and the Structure of its Bands, as photographed at Glasgoiv. By L. Becker, Ph.D., Professor of Astronomy in the University of- Glasgow. (With Three Plates), .......... 251 (Issued separately 94ii September 1904.) XI. The Histology of the Blood of the Larva of Lepidosiren paradoxa. Part I. Structure of the Resting and Dividing Corpuscles. By Thomas H. Bryce, M.A., M.D. (With Five Plates), .......... 291 (Issued separately 19th November 1904.) XII. The Action of Chloroform upon the Heart and Arteries. By E. A. Schafer, F.R.S., and H. J. Scharlieb, M.D., C.M.G., . . . . . . .311 (Issued separately lith December 1904.) XIII. Continuants resolvable into Linear Factors. By Thomas Muir, LL.D., . . . 343 (Issued separately 13th January 1905.) XIV. The Igneous Geology of the Bathgate and Linlithgow Hills. By J. D. Falconer, M.A., B.Sc. (With a Map), ......... 359 (Issued separately 9th June 1905.) XV. On a New Family and Twelve New Species of Rotifera of the Order Bdelloida, collected by the Lake Survey. By James Murray. (With Seven Plates), . . . 367 (Issued separately 3rd March 1905.) XVI. The Eliminant of a Set of General Ternary Quadrics. — (Part III.) By Thomas Muir, LL.D., 387 (Issued separately 15th April 1905.) XVII. Theorems relating to a Generalization of Bessel's Function. By the Rev. F. H. Jackson, R.N., 399 (Issued separately 18th April 1905.) XVIII. On Pennella bal&uopterae : a Crustacean, parasitic on a Firmer Whale, Balajnoptera musculus. By Sir William Turner, K.C.B., D.C.L., F.E.S. (With Four Plates), . 409 (Issued separately 2Qlh May 1905.) XIX. The Histology of the Blood of the Larva of Lepidosiren paradoxa. Part II. Htematogenesis. By Thomas H. Bryce, M.A., M.D. (With Four Plates), . . . .435 (Issued separately 6th May 1905.) XX. Supplement to the Lower Devonian Fishes of Gemunden. By R. H. Traquair, M.D., LL.D., F.R.S. (With Three Plates), .... . 469 (Issued separately 13th May 1905.) EDINBURGH: PUBLISHED BY ROBERT GRANT & SOK 107 PRINCES STREET, AND WILLIAMS & NORGATE, 14 HENRIETTA STREET, CO VENT GARDEN, LONDON. MDCCCCV. Price Twenty-nine Shillings and Sixpence. ( 229 ) IX. — On the Measurement of Stress by Thermal Methods, with an Account of some Experiments on the Influence of Stress on the Thermal Expansion of Metals. By E. Gh Ooker, M.A. (Cantab.), D.Sc. (Edin.), F.RS.E. ; Assistant Professor of Civil Engineering, M'Gill University, Montreal. (With Two Plates.) (MS. received April 11, 1904. Read June 6, 1904. Issued separately September 2, 1904.) CONTENTS. 1. Introduction 2. The Thermal Effect of Tension and Compres sion Stress ...... 3. The Thermal Expansion of Brass and Steel under Tension Stress .... 4. The Behaviour of Iron and Steel under Tensile Stress ....... 5. The Relation of Stress to Strain and Thermal Change in Tension Members PAGE 229 231 232 238 241 PAGE 6. The Relation of Stress to Strain and Thermal Change in Short Compression Members . 243 7. The Variation of Compression Stress in a Long Compression Member ..... 247 8. The Variation of Stress in the Cross Section of a Beam 248 9. Conclusion ....... 250 1. Introduction. In the determination of the effects of stress upon different materials, the investigator has several methods of attack open to him, each of which has its own particular advantages. In the great majority of cases the material under investigation obeys the generalised Hooke's law, and the effects of a stress are therefore most easily followed and measured 'by observations of the strains produced. The strains being usually exceedingly minute, it is necessary to magnify them sufficiently to allow of accurate measurement. To this end many instruments have been devised for measuring the Strains obtained by the action of different stresses, and in fact the great majority of our experimental knowledge has been obtained in this way. The application of polarised light to the determination of stress was first suggested by Brewster,* and he applied it to many problems, particularly the determination of the neutral axis of a glass beam. Neumann,! with a full knowledge of the work of Brewster, developed a theory of the analysis of strain by polarised light, and Maxwell J also independently developed a theory. A third method, which has assumed great prominence in recent years, is the microscopic examination of metals under stress, as developed by Ewing, and Kosenhain,|| and others. The present paper is mainly concerned with the measurement of stress by the temperature changes produced, a subject to which attention was first drawn by Weber,§ who found that when a wire was stretched suddenly a thermal effect was produced, * Trans. R.S.E., vol. iii. t Abhandluugen der k. Akademie der Wissenschaften zu Berlin, 1841. t "On the Equilibrium of Elastic Solids," Trans. R.S.E., 1853. || "On the Crystalline Structure of Metals," Trans. R.S., 1900. § Poggendorffs Annalen, Bd. xx., 1830. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 9). 35 230 PROFESSOR E. G. COKER ON which he attributed to an internal cause. He deduced a theoretical formula in which the change of temperature t was shown to bear a linear relation to the stress p and to the coefficient of expansion Jc, which may be expressed in the form t = c k p, where c is a constant. He proved from his experiments that a wire when stretched within its elastic limit is lowered in temperature, and when compressed has its temperature raised. Duhamel * investigated the modifications which became necessary in Poisson's elasticity equations, when allowance is made for change of temperature. The subject was placed on a sound basis by Lord Kelvin, t who deduced from the laws of thermo-dynamics the general equations of thermo-elasticity. He showed that the thermal effect H produced by stresses p xy can be expressed in the form 3^~dt xv where t is the temperature, J is Joule's equivalent, and e xy is the strain corresponding to the stress p xy . The general conclusions deduced were that " cold is produced whenever a solid is strained by opposing, and heat when it is strained by yielding to any elastic force of its own, the strength of which would diminish if the temperature were raised, but that, on the contrary, heat is produced when a solid is strained against, and cold when it is strained by yielding to, any elastic force of its own, the strength of which would increase if the temperature were raised." These conclusions were experimentally verified by Joule, J who showed that the thermal changes produced by stretching and compressing metals, timber, etc., and by the deflection of helical springs, was proportional to the stress applied, and obeyed the Thomson law. The change of temperature was measured by thermo-electric couples composed of iron and copper wires, either pressed against the specimen or inserted in holes drilled into them ; a galvanometer was placed in circuit with the thermo-j unction to indicate the change of temperature ; to calibrate the galvanometer the test specimen was plunged into water, at a known temperature, to within a short distance of the junction. Edlund § applied the methods of Joule to the determination of the effects of stress on wires, and instead of an iron-copper junction he used crystals of bismuth and antimony cut to a cylindrical form, and the cut ends were pressed against the wire, so that no variation in the thermo-electric power was possible, as might be the case if the natural cleavage plans were used. His results amply verify Joule's earlier work, and he, moreover, obtained an approximately correct value of the mechanical equivalent of * " Memoire sur le calcul des actions moleculaires developpees par les changements de temperature dans le corps solides," — Memoires . . . par divers savans, vol. v., 1838. t " On the Dynamical Theory of Heat," Trans. B.S.E., 1851. X "On some Thermo-Dynamical Properties of Solids," Trans. R.S., 1853. § "Untersuchung uber die bei voluniveranderung fester korper entstehenden warmephanomene, sowie derein verhaltniss zu dabei geleisteten mechanischen arbeit," Pogg. AnnaL, vol. cxiv., 1861, and "Quantitative bestimmung der bei voluniveranderung der metalle enstehenden warmephanomene und die mechanischen warmeaequivalents, unabhiingig von der inneren arbeit des metalls," Pogg. Annal., vol. cxxvi., 1865. THE MEASUREMENT OF STRESS BY THERMAL METHODS. 231 heat by a direct application of the Thomson formula. In a recent valuable paper by Turner * the methods of Joule and Edlund are substantially followed, and a detailed account is given of experiments on tension stress for metal bars of a size needing a modern testing machine for the stressing agent. 2. The Thermal Effect of Tension and Compression Stress. In the present paper the main object of inquiry is tension and compression, and for these stresses it is easy to deduce a simple form of the relation between the change of temperature and the stress from the equations of thermo-dynamics. If I be the length of unit mass of a rod subjected to a compression stress of intensity p, which shortens the bar by an amount dl, and E be the initial intrinsic energy of the bar and dH. the amount of heat developed by the compression, we have dE + p-dl = Jdh- -'{(D,**(fH = J<p f dp + t p -dt \ . . . . . . (1) where Pt is written for the more cumbersome symbol (%-) • Now the alteration of length is a function of the pressure and temperature, and hence we have dl=- dp + -dt. op at Therefore dE = (jp t - P ^) dp + (jt, - v gj) dt a perfect differential, whence d7\ J P<-PcTj) = dp-{ Jt »-Pdt giving j(dt z _dp i \J_l \dp dt/ dt Equation (l) can be written in the form t t p+ t a perfect differential also, from which we obtain _ pt ^ /dtp _ dp,\ = 1 dl t ' \dp dt) Jdt Now if a body be compressed adiabatically t p dt +p t -dp = dt p t t dl 1 or — = -£-; = . dp t v J dt t p * « Thermo-Electric Measurement of Stress," Trans. Amer. Soc. C.E., 1902. 232 PROFESSOR E. G. COKER ON If a be the coefficient of expansion of the bar for unit increase of temperature, we can write this in the form dt_tal dp~Jt p ' and for small changes of pressure and temperature we can write the equation in the form F tal where the sign of At depends upon the signs of a and Ap, since all the other quantities are essentially positive. For metals « is in general positive, and hence a compression stress will raise the temperature, while a tensional stress will lower it. Strictly speaking, the equation only holds for infinitesimal changes of p and t, and it is therefore essential to show what limitations, if any, are to be imposed in its application to bodies under great ranges of stress. The effect of a varying load upon the specific heat of a body has not been determined, so far as 1 am aware, but it is unlikely to differ by an appreciable amount from the specific heat at atmospheric pressure. The coefficient of expansion when the stress is varied may change to a small extent, and experiments were made by Joule * to examine the effect of stress upon the expansion of various timbers, and he found an increase of expansibility with tension. As far as I am aware, there are no experiments showing what effect stress has upon the thermal expansion of metals ; and as the coefficient of expansion enters into the fundamental equation, a special investi- gation was made of the thermal expansion of brass and steel under different tension loads. 3. The Thermal Expansion of Brass and Steel under Tension Stress. The general arrangement of apparatus adopted is shown in fig. 1, in which A is the standard of a small single-lever testing machine provided with a weigh-beam B and shackles C D for securing the test-piece E in position. The loading of the speoimen is effected by suspending dead weights F from the end of the beam, and the maximum load which could be safely applied was 175 pounds. The ratio of the arms of the lever was 20 to 1, and hence the maximum stress obtainable was 3500 pounds. In order to carry the experiments past the elastic limit, it was necessary to have a specimen of very small section ; and on account of the difficulty of maintaining a solid specimen at a uniform temperature, and at the same time observing the change of length, the specimens were chosen of seamless drawn tube, very uniform in diameter, thereby permitting the outside being turned in a lathe to a manageable section. The ends of the tube were soldered into thick ferrules G, having side tubes H for the insertion of thermometers, and inlet and outlet tubes I F were provided, connecting to a pipe system J, in which water could be circulated at any desired temperature. The circulation was effected by a small centrifugal pump K driven by an eleetric * " On some Thermo-Dynamic Properties of Solids," Phil. Trans., 1859. THE MEASUREMENT OF STRESS BY THERMAL METHODS. 233 motor L, and provided with an extension barrel M, heated by a gas flame N. The rotation of the pump caused water to be drawn in at the eye of the pump from the pipe V, and to be discharged through the perforated partition into the encircling heater M. From thence it flowed through the rising pipe J and the specimen, as indicated by arrows. A very vigorous circulation was maintained, and the pipes were thickly lagged, so that there was practically no difference of temperature between the end points of the tube under measurement. The readings of the thermometers were found to be practically coincident at all temperatures, and therefore in the tabulated results one set of temperature readings is omitted. The alteration in the length of M^^ the tube with increase of temperature was determined by aid of a Ewing extensometer of the original pattern. The instrument (fig. 2) consists of a pair of clips A B, secured to the tube by set screws C ; the upper clip carries a frame D, provided with a calibrating screw P and a reading microscope F focussed on the edge of a thick wire W carried by the lower clip. Any alteration in the length of the tube causes a movement of the wire relative to the microscope, and by aid of a glass scale inside the eye- piece the alteration in length can be determined. The dimensions of the instrument were such that a movement of '00002 inch corresponded to one unit of the scale, and the micrometer screw enabled a calibration of the scale to be effected while the instrument was attached to the specimen — a great advantage with an extensometer. The construction of the apparatus permitted a thick layer of lagging K to be applied. It was proposed at first to surround the specimen by a water-jacket, fed from the circulating system, to ensure the temperature of the tube being absolutely uniform throughout, but this was not done, as the lagging was found to be very efficient, and the extra complication did not seem warranted, particularly as only 234 PROFESSOR E. G. COKER ON comparative readings were required. The thermometer bulbs I were inserted in the main tubes, and secured by flexible rubber joints J, # as indicated. In carrying out a test the centrifugal pump was first set in operation, and when the readings of the extensometer became steady, the gas flame was applied to the extension barrel of the pump, and the readings were taken, at intervals, of the temperature and the extension, until a temperature of about 180° Fahr. was reached A new experiment was then begun with a different loading. The testing machine used was not sufficiently powerful to overstrain the tubes used, as these latter were exceptionally hard, owing to their manner of production ; and for the experiments with a permanent overstrain the tubes were taken out and stressed in a Kiehle testing machine of 60,000 pounds capacity. The results of the tests on the brass tube are shown in Table I., from which it will * " On a flexible Joint for securing Thermometer and like Stems and Tubes," E. G. Coker, Physical Review, 1903. THE MEASUREMENT OF STRESS BY THERMAL METHODS. Table I. Brass specimen. — Internal diameter = '378 inches. External diameter = "525 ,, Area =0 - 1042 square inches. 235 Stress in lbs. per sq. inch ) 960 9,597 19,194 28,791 33,590 Load, lbs. . ! 100 1,000 2 000 3,000 3,500 \ Yg-inch overstrain •^-inch overstrain ^-inch overstrain Temp. Fahr. Reads, a Reads, a Reads . A Reads, a Reads, a Reads A Reads . A Reads, a 70 -20 -23 -22 -21 -20 -22 -22 -22 75 20 23 22 21 20 22 22 22 -21 -21 -21 -21 -21 -21 -21 -24 80 41 44 43 42 41 45 43 46 -23 - 22 -21 -24 -21 -22 -23 -21 85 64 66 64 66 62 67 66 67 -23 -21 -22 -19 -20 -19 -21 -25 90 87 87 86 85 82 86 87 92 -22 -21 -20 -21 -20 -23 -22 -21 95 109 108 106 106 102 109 109 113 -22 -21 -21 -21 -19 -21 -22 -21 100 131 129 127 127 121 130 131 134 -22 -21 -23 -21 -19 -21 -22 -22 105 153 150 150 148 140 151 153 156 - 22 -21 -22 - 22 -20 -21 -22 -23 110 175 171 172 170 160 172 175 179 -21 -21 -21 -23 -20 -21 -21 -24 115 196 192 193 193 180 193 196 203 -23 -21 -21 -21 -21 -22 -23 -21 120 219 213 214 214 201 215 219 224 -22 -21 -21 -23 -20 -21 -21 -24 125 241 234 235 237 221 236 240 248 -22 -21 -21 -24 -21 -22 -21 -23 130 263 255 256 261 242 258 261 271 -22 -21 -21 -24 -22 -22 -20 -25 135 285 276 277 285 264 280 281 296 -22 -21 -21 -21 -23 -22 -22 -24 HO 307 297 298 306 287 302 303 320 -21 -21 -22 -18 -24 - 22 -22 -23 145 328 318 320 324 311 324 325 343 -22 -21 -22 -22 -25 -21 -26 -26 150 350 339 342 346 336 345 351 369 -22 -21 - 22 -22 -23 -22 -20 -22 155 372 360 364 368 359 367 371 391 -22 -21 -21 -23 -23 -24 -24 -26 160 394 381 385 391 382 391 395 417 -22 -22 -22 -24 -23 -23 -24 -26 165 416 403 407 415 405 414 419 443 -22 -23 -21 -20 -24 -25 -22 -28 170 438 426 428 435 429 439 441 471 -21 -24 -25 -24 -23 -25; -23 -24 175 459 450 453 459 452 464 464 495 -21 -24 -21 -23 -25 -24 -25 180 480 479 474 482 477 480 489 — 236 PROFESSOR E. G. COKER ON be seen that the extension for all the different loads below the yield point of the material are practically constant. This is shown graphically by fig. 3, in which the readings are plotted as ordinates, with the temperatures as abscissae. The curves for stresses below the yield point are very nearly straight lines, with the exception of No. IV., when a load of 3000 pounds was applied, corresponding to a stress of 28,790 pounds per square inch. We may neglect the small deviation there shown, since at an increased stress of 33,570 pounds per square inch it disappears, and we may assume that the expansion is practically linear. The mean value of the coefficient of expansion for these five different experiments corresponds to a linear expansion of '00001953 for 1° centigrade, the maximum deviation therefrom being slightly less than 1 per cent. For the overstrained tube the coefficient of expansion was greater, the values obtained being as follows :- Table II. Total Permanent Extension. Inches. Coefficients of Expansion per 1° Centigrade. i 16" 1 1 ¥ •00001963 •00002004 •00002121 A similar experiment upon a steel tube having very thin walls was made, and the results are given in Table III. THE MEASUREMENT OF STRESS BY THERMAL METHODS. 237 Table III. Steel specimen. — Internal diameter =0 "406 inches. External diameter =0 '482 inches. Area =0*067 square inches. Stress in lbs. per sq. inch I 1,493 14,926 29,852 35,824 37,157 Load, lbs. I 100 1,000 2,000 J 2,400 ■J-in. overstrain 2,500 £-in. overstrain Temp. Fahr. Beads A Reads. a Reads. a Reads. a Reads. a 70 -12 -13 -11 -12 -12 75 12 13 11 12 12 -14 -13 -12 -12 -12 80 26 26 23 24 24 1 -12 -12 -13 -11 -10 85 38 38 36 35 34 -12 -12 -13 -13 -12 90 50 50 49 48 46 -12 -13 -12 -13 -11 95 62 63 61 61 57 -13 -12 -11 -12 -12 100 75 75 72 73 69 -11 -11 -12 -12 -13 105 86 86 84 85 82 -13 -12 -13 -12 -11 110 99 98 97 97 93 -14 -13 -13 -12 -13 115 113 111 110 109 106 -13 -11 -12 -13 -13 120 126 122 122 122 119 -12 -12 -13 -11 -13 125 138 134 135 133 132 -IS -12 -13 -12 -11 130 151 146 148 145 143 -13 -11 -12 -13 -12 135 164 157 160 158 155 -12 -13 -12 -14 -16 140 176 170 172 172 171 -13 -12 -13 -11 -11 145 189 182 185 183 182 -13 -13 -14 -13 -12 150 202 195 199 196 194 -14 -11 - 12 -12 -14 155 216 206 211 208 208 -12 -12 -13 -12 -13 160 228 218 224 220 221 -13 -12 -14 -15 -13 165 241 230 238 235 234 -13 -13 -12 -15 -13 170 254 243 250 250 247 -16 -13 -15 -14 175 259 263 -13 265 -15 260 180 | — 276 280 — TRANS. RO Y. SOC. EDIN., i IOL. XLI. PART IT. (NO. 9). 36 238 PROFESSOR E. G. COKER ON This differed somewhat from the brass tube previously experimented upon, in showing a practically uniform coefficient of expansion under every load below and above the yield point (fig. 4). The mean value of the coefficient Avas found to be '00001121, the maximum deviation being for the two lowest loads, and amounting to nearly 2*4 per cent., while for the remaining loads the deviation was less than 1 per cent. The general accuracy of the results was checked by comparison with the known values of the coefficient of expansion at atmospheric pressure, and the agreement is sufficiently close to make it clear that the observations were accurately taken, having regard to the fact that the seamless tubes experimented upon had been subjected to exceptional treatment in the process of manufacture, and were probably in a very different physical condition from solid bars of the same material rolled or cast in the ordinary way. We may conclude from these experiments that there is practically no difference ' in the linear expansion of brass and steel within the range of stress up to the yield point of the material, and that for brass there is probably an increase in the coefficient beyond the yield point, but that there is no increase for steel. In this part of the work I was greatly assisted by Mr Charles M'Kergow, Demonstrator in the Civil Engineering Department, M'Gill University, who kindly undertook the major part of the work of observation, and who also rendered me very able assistance in the experimental work detailed below. 4. The Behaviour of Iron and Steel under Tensile Stress. In applying the thermal method of measuring stress, the most convenient arrange- ment of apparatus consists of a thermal junction or pile of the necessary delicacy con- nected to a galvanometer giving a sufficiently wide range of readings for the small difference of temperature produced. In nearly all testing laboratories the presence of iron in large quantities makes it necessary to choose a galvanometer which is not influenced by the proximity of iron, and this was especially necessary in the present case, since in the M'Gill University testing laboratory the main testing machines have, in the course of time, become magnetised, owing to the subsidiary mechanism being operated by electric motors. These difficulties are easily overcome by the use of a D'Arsonval galvanometer, the field of which is very uniform, even in the neighbourhood of large masses of feebly magnetised iron. The galvanometer coil was specially wound for me by Dr Tory, and was of approximately the same resistance as the thermopile used in the majority of the experiments, so that the arrangement was as sensitive as possible. To avoid short-circuiting of the pile when in contact with the metal specimen under test, it is convenient to insulate it therefrom, and I have found a thin sheet of paper, as suggested and used by Joule, the most convenient. The connections of the galvanometer to the thermopile were made by soldered joints, which were afterwards wrapped in paper to insulate them from one another, and then tied together and lagged with cotton -wool. THE MEASUREMENT OF STRESS BY THERMAL METHODS. 239 In making observations with a galvanometer provided with a moving coil of con- siderable weight, such as that of the D'Arsonval type, the indications may not be a faithful record unless certain precautions are observed, for the galvanometer does not take up its position of equilibrium at once, and therefore any error due to this lag will make a considerable difference in the results. It was found experimentally that the time rate of loading could be so determined by trial that the reading of the galvanometer was very approximately a maximum for the stress at any given instant, and by making special experiments for each bar, the rate at which the loading ought to be applied could be easily determined. As an example of the accuracy with which the loading could be applied to keep step with the galvanometer, reference may be made to the results obtained from a steel bar of rectangular section, having a breadth of 0'86 inches and a thickness of 0*315 inches, which was loaded at a uniform rate until a maximum of 4000 pounds was reached, corresponding to a stress of 14,760 pounds per square inch. The readings obtained were as follows : — Table IV. Load, pounds. Time in seconds. Observed Reading. Corrected Reading. 2000 25 4-0 4-23 3000 45 7-0 7-71 4000 60 9-8 11-12 4000 90 9-0 10-82 4000 110 8-0 9-98 4000 135 7-0 913 4000 165 6-0 8-23 4000 200 5-0 7-25 4000 260 4-0 6-34 4000 340 3-0 5-60 4000 410 2-5 4-81 Notes. — Scale distance, 83| inches: 1 division = - 5 inches on scale. Resistance of thermopile and leads = 6 - 07 ohms. Resistance of moving coil of galvanometer = 5*2 ohms. Temperature 68° Fahr. From which it will be seen that as soon as the loading reached a maximum, the readings also attained a maximum, and then began to decrease. The ascending portion of the 240 PROFESSOR E. G. COKER ON curve showing the relation of the thermal change to the stress is, however, influenced by losses due to conduction and radiation, and hence it is necessary to correct for these in order to obtain a correct relation. If it be allowed that the lag of the galvanometer is not a factor in this correction, the required result may be obtained as follows. Let be the diminution of temperature per second due to the application of a stress increasing uniformly with the time, and let 6 be the actual difference of temperature at any time t from the commencement of the application of stress, then 0<0 t, since there is a loss due to conduction and radiation, depending on the difference of temperature between the specimen and surrounding bodies. The loss due to this cause can be very approximately determined by observation of the subsequent readings when the application of stress has ceased, and it was found in all cases that the loss was very accurately proportional to the first power of the difference of temperature. In an interval of time dt, therefore, the diminution of temperature for a tension specimen under uniformly applied stress will be 6 dt — k0'dt, where k is a constant to be determined. The actual decrease of temperature in the time dt is -rrdt. Hence an integrating factor of which is obviously e kt . Hence S'O^oJs'-dt + c k or = o /fc + ce-*'. To determine the constant c we have the condition that is zero at the commencement of the application of the load ; hence c = — 6 /k, and we have or O = &0/(1 -€-*') Now the denominator can be expanded provided the value of the variable t is such that the expression in the bracket remains convergent, and it is evident if kt/2<l i.e. t<C 2 /k this condition will be satisfied. Hence we obtain as a sufficiently near approximation and since & is a very small quantity, this reduces to Now t is the actual decrement of temperature D t due to the stress up to the time t, THE MEASUREMENT OF STRESS BY THERMAL METHODS. 241 and is the observed value D A . Hence the observations for the ascending part of the curve must be corrected by the formula D,= D a (l+|). The value of Jc is determined in each case by the second part of the curve ; and in the example shown and in all others described in this paper, it is of the exponential type. In the present example the value of the deflection D at any time after the loading ceased was found to be - -0045* D = 9-8e where t is the time in seconds from the cessation of the load. The curve showing the readings corrected for the radiation loss during the loading is shown dotted in fig. 5. There is also a correction for the change in resistance of the galvanometer coil and leads, owing to the change in the temperature of the room. The testing laboratory was very favourably situated in this respect, as its temperature rarely varied more than two or three degrees, and hence this correction was unnecessary. A further correction might be made since the current strength i in the thermopile circuit, and therefore the deflection of the galvanometer, =£—5— and where 4> is the angle turned ° COS <p ' ° through by the moving coil, but in all cases the deflection was so small in comparison with the distance of the scale from the moving coil and mirror attached thereto that the correction was negligible. 5. The Relation of Stress to Strain and Thermal Change in Tension Members. The variation of strain with regard to tension stress follows a linear law very approximately over a certain range in the case of most metals, and in the case of iron and steel this linear relation holds for a considerable part of the whole range of stress up to rupture. This is easily shown by delicate extensometers, such as those devised by Unwin, Ewing, Martens and others. It becomes of importance to determine what is the relation of the thermal change to stress and to strain. The only previous experiments of which I am aware are those of Turner,* who has experimented upon the relation of thermal change to stress ; and from the known properties of iron and steel as regards strain, he has deduced from his results that " the thermal limit of proportionality is lower than what is considered the true primitive elastic limit of the metal." He suggests that there exists from the thermal point of view a well-defined range of almost perfect elasticity, beyond which " there is a considerable, in fact nearly equal, range of imperfect. elasticity, before reaching the limits of apparent elasticity of shape." This is a matter of considerable importance in regard to the question of repeated stress, since if this is so, it may have an important bearing on the results of Wohler and others. * hoc. cit.> ante. 242 PROFESSOR E. G. COKER ON In order to test the truth of this, several experiments were made on bars in tension, using a thermopile for measuring the cooling effect caused by the stress applied, and an extensometer to determine the strain. This latter was of the usual Unwin pattern, except that the metal distance-pieces were replaced by mahogany rods, previously soaked in paraffin wax,-' and suitably capped. This precaution renders negligible any error due to any slight changes of temperature, as the coefficient of expansion of mahogany is extremely small. The steel bar quoted above, for which the correction factor for radiation and conduction had already been determined, may be quoted as an illustration. The load was applied as uniformly as possible in a Buckton testing machine at a uniform rate of 4000 pounds per minute, and the galvanometer and extensometer readings were taken at each interval of 1000 pounds. The following readings were obtained : — Table V. Extensometer Observed Corrected Galvanometer Time, seconds. Load, pounds. Reading. Galvanometer Reading. Reading. 7„-=-0045. 16 1,000 2-5 2-59 30 2,000 200 -105 5-0 534 44 3,000 305 -105 8-0 8-79 60 4,000 410 -105 10-5 11-92 75 5,000 520 -105 13-3 15-54 91 6,000 625 -100 16-3 19-60 7,000 725 -100 18-3 120 8,000 825 -102 20-5 26-04 9,000 927 -103 22-8 150 10,000 1,030 -100 25-0 33-44 11,000 1,130 -120 26-8 180 12,000 1,250 28-7 40-32 195 13,000 30-0 43-16 14,000 1 Went off scale at Galvanoniet er reading 13,600 lbs. went off scale Note. — Distance of scale from mirror of galvanometer = 6' 11 \". The galvanometer used in all the experiments mentioned in this paper was provided with a coil of resistance 5'2 ohms, and having about 300 turns; the thermopile was THE MEASUREMENT OF STRESS BY THERMAL METHODS. . 243 approximately square in section, with 31 couples, and had a resistance of 5*55 ohms. The short connecting wires or leads had a resistance of 0"52 ohms. In a few experi- ments, which are specially noted, a linear pile was used of 10 couples and of 0'18 ohms resistance, and also long connecting wires or leads of 1*81 ohms resistance were used in some cases. A plot of these readings is shown on fig. 6, in which curve I shows the relation of the stress to the galvanometer readings. In order to obtain the true reading, correction must be made for the losses due to radiation and change of resistance. The (kt\ 1 + -gj are shown in the table above, and the plot of these, with the load as abscissee, is shown by curve II, giving almost exactly a straight line to near the yield point (fig. 6). The stress- strain relation obtained from the extensometer readings is plotted for comparison upon the same diagram, the unit of extension being 0*00001 inch, and this also exhibits a nearly linear relation up to the yield point. The result of the experiment appears to show that the thermal changes do not indicate a range of imperfect elasticity within the apparent limits of elasticity of shape. A second experiment upon a wrought-iron specimen having a section 2 inches by 0'25 inches was next subjected to stress in the testing machine in a similar manner, and the observations are recorded in Table VI., and a plot of the readings is shown in fig. 7. The observations made to determine the radiation loss are omitted, as they are of a similar character to the example quoted above. The value of k obtained was 0'0031, the time being measured in seconds. The general character of the diagram is the same as in the last case ; there is a gradual bending over of the galvanometer readings towards the time axis, the deviation from a straight line being nearly in a geometrical progression with regard to time. The apparent coincidence of the lower readings with the dotted straight line is probably not exact. It should be noted that the stress-strain curve would practically coincide with the corrected thermal stress curve if sheared over, except near the upper end, where the heating effect begins to play a part. In both cases the thermal readings begin to show deviations from a linear relation to the stress at about the same value of the stress. In other experiments upon different bars of iron and steel, results were obtained confirming those quoted above. It therefore appears probable that the thermal change is very nearly proportional to the stress, in the same manner as the strain ; and that, for the material experimented upon, there appears to be no range of imperfect elasticity as measured by thermal change, coinciding with a part of the range of perfect elasticity as determined by the strain. 6. The Relation of Stress to Strain and Thermal Change in Short Compression Members. It is well known that the relation of stress to strain in short compression members of wrought-iron and steel follows a linear relation for a considerable range of stress, and 244 PROFESSOR E. G. COKER ON Table VI. Time, seconds. Load, pounds. Extensometer Readings. Observed Galvanometer Readings. Corrected Galvanometer Readings. £=•0031. 1,000 -55 10 2,000 65 -60 1-70 1-73 ... 3,000 115 -52 3-30 35 4,000 167 -52 5-20 5-48 45 5,000 219 -53 6-90 7-38 55 6,000 272 -58 8-60 9-38 63 7,000 330 -60 10-30 11-31 75 8,000 390 -48 12-40 13-84 83 9,000 438 -59 13-90 15-69 93 10,000 497 -60 15-50 17-57 103 11,000 557 -58 17-10 19-83 115 12,000 610 -60 18-60 21-91 123 13,000 670 -57 19-10 22-74 132 14,000 727 -58 21-50 25-92 145 15,000 785 -63 23-00 28-17 150 16,000 848 23-90 29-46 155 17,000 17,500 : 24-90 30-88 Note. — Distance of scale from galvometer mirror = 6' 0|". that generally there is no very definite yield point — the strain gradually increasing beyond a certain load, so that the curve showing the relation of stress to strain is well rounded, and therefore the yield point is not so well defined. In order to obtain pure compression stress without bending, it is necessary to keep the specimen as short as possible, and experiments were first made upon compression specimens only long enough to accommodate the thermopile, the strain-measuring apparatus being secured to the compression plates of the testing machine. This arrangement did not give satisfactory results, and after several trials new specimens were prepared, sufficiently long to allow of a strain-measuring instrument being applied to them in addition to the thermopile. The shortest specimen which could be used under these conditions was 4*5 inches long, and, as might be expected, the specimen usually failed by buckling, so that it was not THE MEASUREMENT OF STRESS BY THERMAL METHODS. 245 possible generally to trace the relation of stress to strain and thermal change for pure compression stress up to the point where the departure from Hooke's law was very definitely marked ; but sufficient work was accomplished to show that the strain and thermal change are both proportional to the stress throughout the greater part of the elastic range, and it seems highly probable, from the evidence obtained in the tension experiments, that this will hold throughout the whole elastic range of stress, as deter- mined by strain measurements. In order to indicate the nature of the results the following experiment may be quoted. The specimen was of wrought-iron, 0*9 x 0'39 inches in section and 4 '5 inches long. The strain-measuring instrument used was one specially designed by Professor Ewing for compression, and similar in principle to the extensometer used in a previous experiment, except that the distance between the grips was 1*25 inches, and there was no calibrating screw. The instrument was first calibrated on a Whitworth measuring machine, and the position of the micrometer eye-piece determined, so that one division of the scale corresponded exactly to ^roiWo °f an mcn - The instrument was then set up on the specimen, and the thermopile applied to the broad face. The specimen was, for convenience, stressed in a small press, provided with an hydraulic diaphragm, accurately calibrated up to 21,000 pounds, and the load was applied as uniformly as possible. Preliminary experiments were made to obtain the correction factor for radiation and conduction, and the value of k was found to be '0096. A load was applied at the rate of 2000 pounds in ten seconds, until the specimen failed by buckling. The following readings were obtained in this way : — Table VII. Corrected Time. Load. Conipressometer Reading. Galvanometer Reading. Reading. £=•0096. -58 10 2,000 58 -60 3-2 3 45 20 4,000 118 -59 6-6 723 30 6,000 177 -57 9-8 11-20 40 8,000 234 -60 12-2 14-54 50 10,000 294 -58 15-0 18-60 60 12,000 352 -58 176 22-66 70 14,000 Failed by bending 410 19-6 26-18 Notes. — Long connecting leads. Distance of scale from galvanometer mirror = 10' 2" TRANS. ROY. SOC. EDIN, VOL. XLI. PART II. (NO. 9). 37 246 PROFESSOR E. G. COKER ON The specimen failed by bending at the lower end, due no doubt to the compression plate being provided with a hemispherical seat. The galvanometer readings uncorrected (fig. 8) show a marked deviation from a straight line, but the corrected readings follow a linear law almost exactly, and show no trace of the bending stress, as the thermopile was set against the top of the specimen farthest from the place of failure. The strain readings are plotted on the figure for comparison, and it is evident that the linear correspondence is quite definite. As a further example, the following test may be quoted of a steel specimen l'Ol x # 38 inches in section and 4f inches in length. The value of k found by a preliminary test was '0089, and the test gave the following readings : — Table VIII. Extensomoter Galvanometer Galvanometer Readings. Corrected. k =-0089. Load, pounds. Time, seconds. Reading. Readings. Uncorrected. 1,000 -57 3,000 10 57 -55 2-8 2 93 5,000 20 112 -56 6-0 6-52 7,000 29-5 168 -53 9-3 10-50 9,000 40 221 -60 11-5 1350 11,000 51 281 -62 13-9 17-00 13,000 60 343 -60 16-6 21-00 15,000 70 403 -58 18-5 24-30 17,000 81 461 -65 20-1 27-30 19,000 90 526 22-2 31-20 100 Went off scale 23-5 34-00 110 Notes. — Long connecting leads. Distance of scale from galvanometer mirror = 10' 2". The specimen showed a very slight lateral set near the centre after the application of the stress, showing that the stress had been carried nearly to the point of failure, but this is not apparent in the plot of fig. 9. Other compression tests showed the same general characteristics, and we may therefore conclude that the thermal change accompanying compression stress for iron and steel is linearly related to the stress applied through approximately the same range as the strain. THE MEASUREMENT OF STRESS BY THERMAL METHODS. 247 7. The Variation of Compression Stress in a Long Compression Member. The stresses in any but a short bar are always influenced by bending, and if we assume, in pillars of rectangular section, that the lateral deflection is proportional to the square of the length I, and inversely as the thickness t, it is easy to show, with the usual assumptions of technical elasticity, that the maximum stress in a long column is greater than the stress in a short specimen of the same section, by an amount cl 9 /f, where c is a factor depending upon the fixing of the ends. The value of c for the case of a pillar with fixed or squared ends is often taken as 3-0V ¥> where I and t are expressed in the same units. For technical applications this formula is widely used in the slightly modified form proposed by Rankine, viz. : — p -/I l + el where p is the allowable working stress,/' is the safe working stress in direct com- pression, and r the least radius of gyration of the section, the constant c being adjusted to agree as closely as possible with experimental values. As an example of the ease with which the stresses at different parts of the same column may be compared by thermal methods, the following result on a specimen with squared ends 15 inches long and 1*375 by *625 inches in section may be quoted. The load was applied at the rate of 4000 pounds in 25 seconds, and the galvanometer showed no lag with the rate of loading. The thermopile was a linear one, and it was pressed against the broad face of the specimen in the direction of the breadth. The maximum deflections for different positions of the thermopile were as follows : — Table IX. Espt. 1 2 3 4 5 Distance of Pile from the top end of the specimen in inches. Time of appli- cation of Load, seconds. Load, pounds. Maximum Deflection. Deflection cor- rected to 70 - 4000 lbs. 1-38 3-88 7-50 11-25 13-38 25 25 25 25 25 70-4000 70-4000 70-4000 70-3900 100-4000 2-65 3-00 3-05 2-95 2-90 2'65 3-00 3-05 3-03 2-90 Notes. — Linear pile, long connecting leads. Distance of scale from galvanometer mirror = 10' 5|". As the maximum deflections differed very little from one another, and there was no perceptible lag of the galvanometer, no correction for radiation was necessary. The actual readings obtained are plotted as ordinates on fig. 10, the distances along the bar 248 PROFESSOR E. G. COKER ON being used as abscissae, and the corrected maximum readings of column are plotted in the same way to show the variation of thermal effect. It will be seen that the corrected readings are not symmetrical with regard to the centre of the length of the pillar, so that we may infer that the ends were not in exactly the same condition as regards fixture, and therefore it would be difficult to draw any definite conclusions ; but the experiments serve to demonstrate the value of the method, and it appears probable that further experiments in this direction will be fruitful of results. As a further example of the application of the method, we may quote some experiments on the variation of thermal effect in beams. 8. The Variation of Stress in the Cross Section of a Beam. The assumptions of the Bernouilli-Eulerian hypothesis for beams lead to the simple result that there is a neutral plane perpendicular to the plane of symmetry, and that the stress at any point of the section varies as the bending moment and as the distance from the neutral plane. The assumptions of the above theory have been shown to be false by Pearson,* who has proved that for a beam of circular cross section, subject to a surface load perpendicular to the axis of the beam, the stress does not vary according to the distance from a neutral axis, nor according to the bending moment. The varia- tion of stress at the surface of a beam has been determined by more than one experimenter, chiefly by observations of the strains ; and in order to establish the value of the thermal method for determinations of this kind, a steel I beam was chosen of the section shown in fig. 11, and this was subjected to a uniform bending moment by apply- ing equal loads at two points, each distant 4 inches from the central section of a span of 5 feet. In this way the bending moment at the central section was made as uniform as possible. The thermopile was pressed against the beam at five different places in succession, and the deflections of the galvanometer were noted for approximately the same loading applied at a uniform rate. The value of the correction factor for each experiment was determined in the usual manner, and its value was found to be very constant, except in the last set of readings. The observed and corrected readings are plotted in place upon fig. 11, and from them a curve has been drawn, the ordinates of which represent to a reduced scale the maximum readings for a total load of 5000 pounds. The variation of thermal change is seen to be proportional to the distance from a point slightly above the centre line, and (fig. 11) to obey a linear law almost exactly. These results agree in general with those obtained by Professor Bovey, F.R.S.,t who used a very delicate roller extensometer. He found an approximately linear relation for the strains, and in most cases the neutral axis was somewhat above the centre of gravity of the section towards the compression side, a result which may be expected, having regard to the probable distortion of the section by the bending moment. The * " On the Flexure of Heavy Beams subjected to Continuous Systems of Load," Quart. Jour. Math., 1889. + " A New Extensometer," Trans. Roy. Soc. Canada, 1901. THE MEASUREMENT OF STRESS BY THERMAL METHODS. 249 Table X. Time, Thermopile on top of Beam. 4" above centre. k= -0044,5000 lbs. in 60 seconds. Thermopile T72 above centre. k = •0046, 4950 lbs. in 60 seconds. Thermopile T V' below centre. Thermopile 1-81" below centre. k = ■0044, 5000 lbs. in 60 seconds. Thermopile 4" below centre. £="0054, 5080 lbs. in 60 seconds. seconds. GalV. Reads. Uorr. do. GalV. Corr. Reads. do. GalV. Corr. Reads. do. GalV. Corr. Reads. do. Galv r . Corr. Reads. do. 10 1-30 1-53 •65 -66 - -85 - -87 -1-5 -T54 20 2-50 2-61 1-75 1-83 -•03 -3-5 -3-70 30 3-55 3-78 2-38 2-54 -•07 -2T5 -2-3 -5-8 -627 40 4-90 5-33 2-75 3-0 -TO -2-50 -272 -6-5 -7-20 50 5-80 6-43 3-00 3-35 -T2 -2-75 -3-05 -7T -8-06 60 7-00 7-92 3-15 3-59 -14 -310 -351 -7-7 -8-95 75 7-60 8-85 308 3-61 -•14 -3-30 -3-84 -7-7 -9T0 90 7-40 8-87 2-90 3-50 -14 -3-25 -3-89 -7T -8-8 105 ... - 3-00 - 3-69 Table XL Distance of pile from centre line of beam in inches. Load, pounds. Corrected Reading for Radiation. Corrected to 5000 lbs. 4-00 5000 8-87 8-87 172 4950 3-61 363 - -063 5000 - T4 - 14 -1-81 5000 -3-89 -3-89 -4-00 5080 -9T0 -9-09 results obtained were confirmed for the same loading by other tests. In some experi- ments on the variation of stress in cement beams two months old with steel reinforcing, the thermal method did not give satisfactory results, and apparently the combination does not appear to behave like a true elastic solid. On the other hand, specimens of cement of considerable age behave exactly like iron and steel. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 9). 38 250 THE MEASUREMENT OF STRESS BY THERMAL METHODS. Conclusion. The application of the thermopile to determine the thermal change in a body, and I hence the intensity of the stress, has an extremely wide application, since the thermo- elastic equations deduced by Lord Kelvin are generally applicable to elastic bodies subjected to every type of stress. Only a limited number of cases have been treated in the preceding pages, chiefly with a view of showing the range of application of the method. The thermopile, while probably forming the most convenient method of determining the thermal state of a body under stress, is not the only one which could be applied. During the summer of 1903, the author, by the kindness of Professor Cox, Director of the Physics Building, M'Gill University, was enabled to experiment with a Boys radio-micrometer, which was set up in close proximity to specimens in tension and compression, and the readings were found to be proportional to the load ; but, with the limited experience of the author, it was found to be much less convenient than a thermo- pile, mainly because of the extreme delicacy of the apparatus, and the difficulty of setting it horizontally upon a steady base near the testing machine. In conclusion, the author desires to express his warmest thanks to Professor Bovey, F.R.S., for the use of his well-equipped testing laboratory, and to Professor Ruther- ford, F.R.S., for valuable suggestions during the progress of the work; also to Mr M'Kergow, Demonstrator in Civil Engineering, for his untiring assistance in making observations. s R /. Soc. Edin. Vol. XLI. Coker: On the Measurement of Stress by Thermal Methods. — plate i. oo too no llo , /3c /to ISO t 160 no 180 3&mJ> re FnhZ 1° So a too no mo I3c ;*> lyo /So 170 180 KxhZ. td&om. ' 36o 7Vm£, Seccmds 10 xo 3o q<, so «3o Jo GnLuKH/namdfcr ftcauiZno, a 2o° foo coo Soo Mo lxo lf „ E^tvnso^luJ -BeaM^s oy. Soc. Ed in. Coker: On the Measurement of Stress by Thermal Methods. — plate ii. I jsncie Vol. XLI. 2? foo, le loo, f- v\ f / J / / s- / / ' / / . / / 1 J / / / s / ' / i / / / // '/ / / / / / I &al rafiwmetor jr do ReacUnqs Conedvob JttSfcrees-sCnxw-v auAvei O 2o ^O <oo 8o Ti/TLe Sec<7ruis tyooo 8ooo iZooo ibooo LoixcL fxnjsmdLs f d s 'o /Z /f- /6 &o Hf- GoJbttyuTixirn£t£rr Heaaji loo fco Geo 8oo /coo /too Vruoim, Ext&nsomeitcf 3.0 "— F5 - -i Tv — -~y \ ^ \ v- A ""*? h a-o Fin » / / / J ; 00 r il ) A "-- L -| 1 ob O 1-35 S-9fS r-5" //•2^' /3-3ff" 15 Bottom; i ( 251 ) X. — On the Spectrum of Nova Persei and the Structure of its Bands, as photo- graphed at Glasgow. By L. Becker, Ph.D., Professor of Astronomy in the University of Glasgow. (With Three Plates.) (MS. received May 2, 1904. Read June 6, 1904. Issued separately September 9, 1904.) The spectrum of the new star in Perseus, which Dr Anderson, of Edinburgh, discovered 1901 February 21, was photographed at the Glasgow Observatory from 1901 March 3 till 1903 January. From the early photographs one gains the impression that the spectrum consists of a number of bright bands of different lengths, fading towards the ends, and overlapping each other, thus producing a series of maxima and minima of brightness. Near wave-length 5000 the intensity rapidly falls off towards the less refrangible side, and the bands appear detached. The middle of each of the three most intense maxima approximately coincide with the hydrogen lines H^ , H y , H s , and on two photo-plates, March 18 to 20 and March 25, each of the bands is crossed by a sharp Fraunhofer line. On the photo-plates taken after 1901 August 1 the bands are all detached ; some, including the two bands whose middles approximately coincide with the principal nebular lines, have almost the same lengths, and suggest a line spectrum in which the lines have been broadened into bands, others are considerably longer and have pronounced maxima. While it was probable that the three hydrogen lines and the two principal nebular lines were represented in the spectrum by bands, it remained to be proved that the wave-length of a definite point of the band bore a definite relation to the wave-length of the line to which it belonged. As a result of my investigations, founded on micro- metric measurements and estimates of intensity, I shall show that the bands which con- tain a series of reversals are similar in type, and that the ratio of the distance between any two points in a band to that between corresponding points in another band is the ratio between the wave-lengths. The spectrum of Nova Aurigae resembled that of Nova Persei very closely ; its changes followed the same course, and it showed the con- siderable broadening of the lines into bands, the structure of which has, however, never been investigated. The systematic broadening of the spectral lines into bands, which for Nova Persei amounted to a 35th of the wave-length in March and April, and a 100th after August, seems to be a feature of new stars, and ought to be accounted for in an explanation of these objects. 2. The Spectrograph. — The spectrograph of 8 cm. aperture is connected to the Breadalbane reflector of 51 cm. aperture and 446 cm. focal length. The equatorial mounting of this instrument, probably made by the late Thomas Grubb some fifty years TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 10). 39 252 PROFESSOR L. BECKER ON ago, is remarkable in so far as its inclined stand anticipates the chief structural feature of the Potsdam astrographic refractor. The reflector, which had proved useless in its old condition, obtained in 1895 a new silver-on -glass mirror, driving clock with electric control, driving sector and declination clamp with slow motion, all from the works of Sir Howard Grubb. The mounting of the spectrograph was supplied by a local smith. A platform forming a right-angled triangle extends from the upper end of the tube to the free end of the declination axis, and its plane is inclined 35 degrees to the optical axis. Parallel to it, the central ray of the reflector is reflected by a plane mirror. The platform is a stiff structure for its weight. It consists of two layers of corrugated iron, with the corrugations crossed and bolted at every point of contact, and it is strengthened by thin sheet steel ribs. To it is clamped a quarter-inch steel sole- plate, with adjustable bearings for the two tubes of the spectrograph, and on this sole-plate a small cast-iron table carrying the prism-box can be adjusted and clamped. The platform rests at its upper end, a corner of the triangle, on a casting which is bolted to the tube of the reflector ; at its lower end, the shortest side of the triangle, it is screwed to a strong cast-iron arm, which is fixed to the declination axis, in place of the balancing weights, at right angles to this axis and the axis of the tube. As I had the declination axis lengthened, and the telescope tube shortened and placed more favourably in its cradle, the movable part of the instrument weighs now less than in its old condition. The object-glass of the collimator has an aperture of 8 "2 cm. and focal length of 74 cm. ; that of the camera, a Cooke triplet, 8 '9 cm. by 149 cm. The focal length of both combined has a large temperature coefficient, 0'13 mm. for a degree centigrade. The prism made by Hilger of white Jena flint glass measures 16*5 cm. on a side, and is 9 "5 cm. in height. Since it was re-annealed its separating power is most satisfactory. The central portion of the spectrograph is enclosed in a box, and by means of a small heating apparatus the temperature of the prism and the object-glasses can, at least to some extent, be kept under control. Unfortunately, the instrument cannot be used in summer after a sunny clay, because in the iron dome, the large prism is heated in such a way that the definition becomes too bad for accurate work. The jaws of the slit are formed by the two halves of a circular mirror 2 - 5 cm. in diameter, and they open symmetrically 0*15 mm. for a revolution of the screw. The width here employed was usually 0'018 mm. The plane of the mirrors which form the jaws of the slit is inclined 7 degrees to the plane normal to the central ray. If the image of the star does not fall on the slit, the rays are reflected towards a small mirror which is fixed to the telescope tube, and thence towards a viewing "telescope" (which is focussed on the slit) of 7 cm. aperture and 30 cm. focal length (two object-glasses mounted close together). It lies almost parallel to the collimator. Owing to the large size of the jaws of the slit, the effective field is half a degree, which is a great con- venience in finding a star and setting it on the slit. The spark apparatus is hinged to the platform in front of the slit. When turned into position, the optical axis of its lens coincides with that of the collimator. THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 253 The collimator lens, though a fine object-glass, F/9, did not prove to be achro- matised for photographic rays ; the focal curve of the spectrograph runs along zero from D to Up, then gradually rises, and at wave-length 3700 the ordinate is 2'8 cm. I had therefore to incline the photographic plate up to 30°. When the telescope is turned through twenty-four hours of hour-angle, the image in the focal plane of the spectrograph oscillates in simple harmonic motion, with an amplitude of 0'8 mm., along a line which is slightly inclined to the spectral lines. 1 compensated this deficiency of the mounting by making the plate-holder adjustable in the direction of the spectral lines, and at right angles to them, the motion being effected by two micrometer screws of '5 mm. pitch. The plate-holder takes the plate, 11 by 4 cm., in its upper half, which lies central in the camera, and in the space below it carries an eye-piece with stout cross wires. As fiducial line for keeping the plate stationary with regard to the spectrum, I employed the magnesium line 4481, which is almost covered by the stout wire in the eye-piece. A split spring, which presses against the jaws of the slit, cuts out a short line from the upper portion of the slit 1"5 cm. above the optical axis, and the magnesium terminals are placed close to it, inside a short glass tube, to guard the slit against tarnishing. By these means I am able to keep a line of the spectrum always on the same place of the photographic plate, and to replace the plate after days in its old position. The differential change of dispersion due to changes of temperature is, of course, not taken into account. During an exposure of the plate I moved the plate-holder every time 0*01 revolution of the micrometer screws, at intervals given by a table, and checked the position once an hour direct on the magnesium line. To illustrate the efficiency of this method, I mention that on photo- plate No. 23, comparison lines at a distance of 0'04 mm. appear separated, though they were exposed on twelve different occasions, five seconds each time, on two days, and at different hour-angles. At the time the new star was announced, wave-length 5200 t.m. was in the centre of the field of the camera, and 4000 at the end of the plate. No change was made in the position of the camera until the beginning of October 1901, when wave-length 4170' was placed, in the centre. The distance of the hydrogen lines H^ and H v is 20 mm. on the plate, and one tenth- metre or Angstrom unit is represented on the plate by 0"1 mm. at A = 3500, 0"05 mm. at X = 4300, and 0'025 mm. at X = 5200 t.m. 3. The Measurements and their Reduction to Wave- Lengths. — The plates taken in March and April 1901, and again those after January 1902, were difficult to measure, — the former, owing to the gradual change of intensity of the spectrum, which presented few definite points to set on ; the latter, owing to the faintness of the spectrum, some parts of which could barely be distinguished from the accidental markings on the film. I finally adopted the rule to measure every point to which the eye was drawn, except those which I thought to be defects in the film. With respect to these, I became more careful as the work advanced ; and it is possible that the earlier plates may contain 254 PROFESSOR L. BECKER ON more than were actually measured of the minima, or reversals, which were present in the spectrum during the whole period. On the plates taken between August 1901 and January 1902 the structure of the bands, including the minima, is easily seen, and in some bands it is visible to the unaided eye. The intensity of the spectrum between every two points measured was estimated on an arbitrary scale, the estimated " degrees" of intensity increasing with the intensity. In this paper the " intensity of the spectrum" stands for the intensity of blackness on the negative, while "intensity of radiation " is used for the intensity of light in the focal plane of the spectrograph. I measured each plate about four times, alternately in opposite directions. Each series includes a number of settings on the lines of the comparison spectrum, iron-calcium until September, and iron-titanium afterwards. The points measured on the same plate were then identified by a graphical process, and all those were discarded which had not been repeatedly observed. Only on plate No. 5 I made an exception, where, after the discussion was finished, I included two minima which had only once been measured. Since the measuring occupied about half a year, and no measurements were taken after the reductions were begun, the results of the different plates may be con- sidered independent. For each plate I reduced each series of measurements separately to wave-lengths, and then combined them to mean values. The tables used in the reductions give the position of the micrometer screw of the measuring apparatus, re- ferred to an arbitrary zero, for each wave-length at an interval of 1 tenth-metre ; they are based on Ketteler's formula of dispersion,* and were prepared for the angles of in- clination at which the plates were exposed. The comparison spectrum determines the correction curve of the zero of the table. The Spectrum in March and April 1901. 4. Results of Measurements. — The results derived from the measurements made on the photo-plates Nos. 1 to 7 are given in Table I. The first column contains the mean wave-length, the second the average difference of one measurement from the adopted mean value, and the third, under the heading " Intensity," the estimated degrees of intensity. The notation |? indicates that the intensity gradually changes from degree 9 to 7. * Annalen der Physik unci Chemie, 1881, 12. THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 255 Table I. Photo-Plate No. 1, 1901 March 3. Angle of inclination of plate = 0° ; width of slit 0-018 mm. ; orthochromatic plate ; exposure 2*4 hours ; the spectrum was measured 6 times and referred to 50 standard lines ; good definition of spectrum ; the more refrangible end beginning at A. = 4500 is increasingly out of focus. A V. Intensity. A v. Intensity. A V. Intensity. A V. Intensity. 3967 5 4317-7 0-5 4 4844-4 0-7 9 5007 1-6 3 3972 1-3 6 4320-2 0-6 3 4847-6 0-8 •\ 5024 6 3 4010 2-4 4321-8 0-4 6 4850-3 1-5 = 15 5046 2 1 5 6 0-3 4049 1-2 4 4325-5 0-3 8 4851-6 1-0 ►14 = 15 5121 2 1 4077-8 0-8 2 4329-3 0-6 12 4852-9 05 = 15 5143 2 4086 2 1-1 4334-1 0-8 4854-8 1-2 ) 5160 1 5 11 12 3 4090-5 0-8 10 43597 0-7 10 4869 1-4 8 5210 1-6 1-5 4097-7 0-4 9 4367 1-5 9 4873-2 0-5 14 5241 3 0-5 4117-9 0-4 9 4412-8 0-8 7 4883-2 0-7 8 5271 1-6 1 4123-7 0-8 c 7 4435 2 6 4890-0 0-8 6 5306 6 3 3 4145-6 0-6 O 4454-2 1-0 7 4896-9 0-5 3 5347 2 3 5 7 2-5 4161-3 1-2 7 4476 * 1-7 9 4902 1 2-5 5362 1 1-5 4186-5 0-7 6 4584 3 6 10 4907 1-3 5 5 5381 2 2 4200-8 0-9 5 4619 4 7-5 4925-0 0-7 4 3 5520 2 4 4209 1-6 4 4640 7 8 4934-2 5544 9 4-5 4221-6 0-5 c 5 4686 3 7 4943-1 0-8 2 ; 5586 1 3 4 4253-6 1-0 O 4 5 4 4801 2 5 6 4959 4 1 5678 3 2 4281 5 4835-2 0-5 4 4981 2 0-3 5711 4 1 4312-5 0-5 4839-3 0-9 3 4985 8 1 5747 3 0-3 4317-7 0-5 4844-4 0-7 8 5007 1-6 2 * The enlargement shows a maximum within this space. Photo-Plate No. 2, 1901 March 6 and 8. Angle of inclination of plate = 0°; width of slit 0'018 mm.; orthochromatic plate; exposure l - 8 ours, 1 h. on March 6 and 0*8 h. on March 8 ; the plate was intensified and is slightly fogged ; the spectrum is not so dark as on No. 1 ; it was measured 4 times, 3 times on the intensified plate, and referred ;o 22 standard lines; good definition; focus same as on No. 1. 4089 4118-2 4296 4319 4324-6 4328-6 4337-8 4347 4355-5 4359-2 v. I Intensity. 2 0-3 2 1-5 0-4 0-4 2 0-6 0-4 A V. 4359-2 0-4 4365-7 0-7 4375-6 1-3 4392-0 0-2 4413-1 0-4 4466-0 0-7 4477 5 4499 4 4528-0 4545 3 Intensity. 1 2 2 3 3-5 3 4545 4600 4616-7 4643-6 4691 4720-2 4840 4846-4 4849-9 4852-3 3 1-8 0-4 0-6 2 1-1 3 0-7 0-7 Intensity. 4 3 4 1 2 7 10 4852-3 4855-4 4857-7 4861-8 4870 4879-7 4881-8 4883-3 4892 4928 0-9 0-3 0-4 2 0-9 1-1 0-1 2 3 Intensity. 10 7 10 8 10 7 0-5 Table I. is continued at the top of the next four pages. 256 PROFESSOR L. BECKER ON Table I. — continued. Photo-Plate No. 3, 1901 March 18 and 20. Angle of inclination of plate = 13°; width of slit 0-018 mm.; Imperial plate; exposure 4*4 hours, T2 h. on March 18 and 3 - 2 h. on March 20; clear negative; the spectrum was measured 6 times, the H-bands 12 times, and referred to 50 standard lines ; good definition. A V. Intensity. A V. Intensity. A v. Intensity. A V. Intensity. 4028 1-5 4251 4 7 4512 4 4871-0 = 9 1 10-5 13 4034-8 0-4 1-5 4260 4 6 6 4535 2 10 4875-1 = 9 13 4037-3 1 9 2 4284-7 0-8 7 4551 2 10 4879-4 11 4044-0 4315-31 = 4562 4 11 4883-9 8 3 7 11 4055 2 4318-4 \ 9 4590 2 10 4895 2 7 3-J 6 4061 3 4 ( 4319-03 4602 3 9 4900 3 4080-76 4319-6 4607-2 o-i 8 4911-4 3 4 4081-51 4326-9 4611 1-3 9 4922 11 3-5 4082-2 6 4335-6 12 4620 3 10 10 4931-2 0-2 3 4085-7 7 4340-9 13 4643 6 11 4941-0 0-5 2 4090-4 4343 14 4679 8 10 4956 2 1 8 14 1 4097-8 9 4348-7 14 = 10 4700 3 8 4978 1 1-5 4102-7 10 10 4358-0 12 4735 7 6 5003 4 o 4117-9 10 4368-9 10 4794 7 5 5 5025 2 A 1-5 4121-1 9 4382 5 9 4833-46 = 5042 3 1 9 5 0-5 4132 5 6 6 4396 1-4 10 4836-5 ) 507- 6 0-2 4148 6 4423 3 10 4837-43 ° 509- 10 0-3 4154-5 11 5 4427-8 0-6 9 4838-4 ! 5133 5 9 8 4165 2 6 4442 2 9 4844-1 10 5154 0-2 4187 3 5-£ 4459 3 10-5 10 4849-4 13 5175 0-2 4203-9 0-7 4473 3 4855-8 5188 0-5 5 11 9 o-i 4219 1-3 6 4492 3 10-5 4858-0 13 5207 o-i 4233 1-6 4502 6 4864-1 = 9 5265 7 10 13 o-i 4251 4 4512 4 4871-0 = 9 5295 5. The three H-bands. — On photo-plates Nos. 3 and 4 three well-defined Fraunhofer lines are a prominent feature, appearing respectively in the neighbourhood of the hydrogen lines H^ H y H 5 , and the difference between the wave-length of a Fraunhofer line and that of a corresponding H-line is proportional to the wave-length. Towards the less refrangible side of the Fraunhofer lines there are bright bands, which also occur on the other five photo-plates. If the wave-lengths of corresponding points of the three bands be compared with the wave-lengths of the three hydrogen lines, the differences are found also to be proportional to the wave-lengths. I conclude that the THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 257 Table I.— continued. Photo-Plate No. 4, 1901 March 25. Angle of inclination of plate =13°; width of slit - 020 mm.; Imperial plate ; exposure 4 hours ; the plate is fogged and the spectrum is much fainter than on plate No. 3 ; the spectrum was measured 4 times and referred to 30 standard lines ; the definition is not so good as on plates Nos. 1, 2, 3, 5. A V. 1 Intensity. A. V. Intensity. A v. Intensity. A v. Intensity. 4041-5 o-i 4318-7 0-2 \° 4540 4 4 4851-8 0-9 0-2 8 4049 1-4 0-5 4320-0 0-6 ) 3-5 4559-7 0-3 5-5 3 4858-0 3 4056-4 0-2 1 4325 1-3 4-5 4589 1-3 3 4859-6 8 4065-0 0-5 o 4328-9 0-4 5-5 4613 2 3-5 4863-4 0-7 ) 4072-6 0-7 3 4335-2 3 4617 1-1 6 4 4865-0 r 4081-1 0-4 4337-8 7 4626-6 0-8 8 4866-1 0-4 8 4083-4. 1-1 4 4344-9 0-4 8 4648-4 0-7 7 4871-0 4092-5 10 5-5 4354-0 4660 2 5 6 4872-8 7-5 4100-8 0-9 6 4354-6 7 4669-1 0-6 6 4879-9 0-7 4 4109 1-6 5 4360-0 0-3 6 4682 4 8 4882 1 4115-6 1-0 4 4365-2 0-3 5 4691-0 0-5 7 4884 3 4119 2 3 4372 1-2 4 4701 3 6 4885-5 0-7 2 4122-0 1-0 2 4384 17 3-5 4722-0 0-8 4 4894 I i 1-5 4135 1-8 4398 5 3 4729 1-8 3 4905-7 0-6 l 1 2 2-5 4157 3 4428-7 2-5 4739-9 0-4 2 4912 1-2 4165 1-6 1 4434 1-3 3 4750 1-1 1-5 4929 1 4207 5 2 4454-6 1-0 4 4770 4 0-3 1 4947 1-1 0-3 4261 1-6 4459-3 0-7 5 4826-3 0-9 4968 1-5 1 6 2 o-i 4277 1-7 4480-5 0-4 4834-4 1-1 1 4986-2 0-2 1-5 5 0-3 4282-8 0-8 4493 4 4836-1 ° 5006 6 2-5 3 0-5 4307 1-7 3 4503-9 0-8 4 4837-9 o-i ) 3 5020 4 0-4 4317-7 0-2 L 4527 3 4-5 4846-2 0-6 5 5045-8 0-8 4318-7 0-2 r 4540 4 4851-8 0-9 bands are due to hydrogen radiations, which, under ordinary conditions, would produce the three hydrogen lines. Tables II, III., IV. prove the statement. Let X be the wave-length of a hydrogen line, A the observed wave-length of a point in the band, s the correction for the orbital motion of the earth, then (1) .... X X + s = A + a. 4500 determines a , belonging to a certain point of a band, and it must be shown that a has the same value for corresponding points of all three bands. In Table II. the values of a are compiled. The positions of the first, third, fourth and fifth minima agree as 258 PROFESSOR L. BECKER ON Table I. — continued. Photo-Plate No. 5, 1901 March 25. Angle of inclination of plate = 13° ; width of slit 0*020 mm. ; Imperial plate ; exposure 3 "2 hours ; the plate is badly fogged ; the spectrum was measured 5 times and referred to 35 standard lines ; good definition. 4032-9 4038-2 4054-4 4062-7 4082 4089 4099-1 4106 4113 4117-3 4124-6 4129 4145 4163-5 4179 42209 4235-0 0-3 0-5 0-8 1-0 1-8 1-2 0-4 1-3 1-3 0-4 0-2 1-2 1-7 0-5 1-3 o-i 0-8 Intensity. A 42350 0-5 4248 1 4279 1-5 42984 2-5 4 4313-8 6 4325-2 * 7 43269 ^ tnin. 4336-5 ) tnin. 4337-3 4343-5 7 6 4345-9 5 4353-3 4-5 4358-4 4 4362-0 5 4364-5 3-5 4366-5 4 4377 0-8 1-1 3 0-4 0-5 0-5 0-4 1-0 0-3 0-5 0-6 Intensity. 5 45 4 5 3 8-5 6 9 7 10 9 8-5 8-5 4 = 4 4377 4405-5 4427-2 4454 4471-3 4477-3 4486-3 4507 4556 4582 4605-7 4615-4 4660 4697-8 4724-7 4753-8 4806 4825-5 5 0-9 0-6 1-2 0-8 0-8 0-9 2 4 4 0-5 06 3 0-5 0-8 1-0 2 0-7 Intensity. 7 6 7 8 8-5 9 7-5 3-5 3 7 6-5 5-5 5 4 A V. Intensity. 4825-5 0-7 6 4839-9 0-2 8 4846-0 4848-8 0-5 11 4855-8 0-8 max. 11 4867-9 0-8 max. 11 4874-5 0-8 9 4878 2 7 4887 1-4 4 2 4898-9 1-0 0-5 4910-1 1-0 9 4932 1-7 1-5 4941-6 0-7 05 4950-6 0-9 1 4963 2 0-5 5037 1-6 0-2 5090-7 10 * One measurement. closely as may be expected from the accuracy of the measurements. The sixth minimum appears as such only in two bands on photo-plates Nos. 5 and 6, but each being measured repeatedly and independently, there can be no doubt that it really exists ; besides, the other five settings made at this point appear to me to suggest that it existed also at those places, though it was not appreciated. The second minimum is questionable, because both positions rest on only one measurement. The estimated degrees of intensity at corresponding points of the same H-band do not agree on the seven plates. The observations are too few to determine the reductions to an average scale of the estimated degrees of intensity as a function of the degrees ; but since the differences of the estimates made on two plates at corresponding places are almost the same for all degrees, I reduce the estimates to the scale employed on photo-plate No. 3 by adding to the degrees on plates Nos. 1, 2, 4, 5, 6 respectively 0, 4, 5, 2, 5. The result is given in Table III. THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 259 Table I. — continued. Photo-Plate No. 6, 1901 April 1 and 5. Angle of inclination of plate = 0°; width of slit O020 mm. ; orthochromatic plate; exposure 6 hours,. 3-2 h. on April 1 and 2 - 8 h. on April 5; plate was intensified and the spectrum is faint; the spectrum was measured 4 times and referred to 30 standard lines; fair definition; focus same as on plate No. 1. A v. Intensity. A V. Intensity. A v. Intensity. A V. Intensity. 4087 1 0-5 4442 4 3 4731 1-6 0-5 4933 1-3 0-5 4101 % 0-8 4462-2 0-7 3-5 476- 8 0-3 4943 3 4114 1 4483 1-3 3 4800 3 1 5037 6 1? 4310-7 0-1 0-3 4498 3 2-5 4846-5 0-6 5 2 5063 3 4328-4 0-8 3 4512 1 * 4853 1-8 7 5160 2 0-2? 4337-5 1-fi = 4 4563 1 4863-8 0-6 5177-0 o-i 5 3 4344 1-8 7 4568 3 3-5 4865-1 0-6 8 5299 6 0-2 4350 1-6 = 6 4593 3 q 4870-7 o-i 9 = 538'S 4 0-5 43559 0-5 4 4614 2 3-5 4876-1 1-0 5 5447 7 1 4358-8 0-9 2 4638 4 4 4878-9 0-9 4 5498 7 0-3 4370 2 t 4663-1 0-8 4882 2 3 5578 2 0-2 3-5 1 4385-1 0-2 0-5 4678-1 0-8 3 4895 4 0-7 0-2 5604 3 0-5 4402-4 0-5 2 4690-0 0-6 2-5 4911 4 0-5 5677-7 o-i 0-8 44305 0-4 2-5 4709-4 0-7 1 4925 1-7 5713 1-7 0-2 4442 4 4731 1-6 4933 1-3 5739 4 * Defect in film 1 Photo-Plate No. 7, 1901 April 10 to May 3. Angle of inclination of plate = 0° ; width of slit 0-020 mm. ; orthochromatic plate ; exposure, near horizon, 9 hours on 7 days ; the plate is fogged ; the spectrum was measured twice and is very faint ; bad definition. A Intensity. A Intensity. A Intensity. A Intensity. A Intensity. 4325 4360 3 4455 4504 1 3 4614 4718 3 4845 4882 3 5005 5021 1 4536 In Table IV. I have combined the results obtained from three plates to mean values, in order to exhibit the agreement in position of corresponding points of the three H- bands as reduced to X = 4500, and also to show in what manner the observed degrees differ. Towards the less refrangible end H Y declines in intensity 4 degrees on 25 tenth- metres and passes into a bright spectrum, while H^ decreases 8 degrees and fades into TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 10). 40 260 PROFESSOR L. 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IN CN CO CO 1-1 OS CN 1 -n CN 1 i— I CM 1 to 1 CM 1 OS + CN + to i— I iH CO iH 1 + CM lO CO CO in t- t^ 0O as" in CO CN (N CO CO ^ to CM OS CM 1 1 1 1 + T + + + t-~ -# O o in -tH CO m 00 m o ■tn o CN CO CO CN o CM o ^ J^ ^x CO CI to CO a 0) 6 1 1 1 1 1 CO 1 + + + + + ■"* os m 00 o f~ r^ CO CO 00 to CO in CO CN CN <N r-H CO CM r-l n Tf o r~ o CM CN CO as 13 1 1 1 1 1 1 + + + + CN ib 1 + CO to 00 as CM to I— I to CN 1^ I— I CN tl( t^ CO CM 1 1 1 1 + + •;u Pd J° " °N H CN CO Tt< in co is 00 as o i-l CM CO •* in to l-^ 00 3S o CI CM "55 a CO as m CD a a 3 a s g a 3 a 1 >> ' HI a as ■4-3 H-C OS 05 cS CD o OS 3 s cm' CO ■* in to" THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 261 § rO "fe. -to O 8 s o «> |- 1— 1 Sf> so < ' CO CO to H 3 <fe CO ^> <3C3 ■to * cO * «o 1 CO CO ^ O <M : cm us CO us -F O OS 00 CO CO us us as '" H 1-1 1-1 V ^H V 1-1 1-1 tS us s US US US 00 00 O o CM CO CO : co : CO CO i— 1 OS CD 'Hi CM CM Cr-I O c& W 3 US V "■■ W O CO 00 o CO CO CO W CO us CO CO OS 00 t^ CD CD CD CO 1-1 1-1 ' -l rH 1-1 ^ X US US US O 00 oo o CO OS CO OS CO OS CO OS CO rH OS CI l^ CO T3 1-1 '~ l 1-1 1-1 rH t-H rH C CM — ».. ■"■ ■»"■ to CO - rH : rH : <M ■^1 tH rH CO US us US us US US r* CO CO 00 OS Ttl : cm : ~1 00 ■^1 rH rH CO CD CO rH i—t CM CM t^ CO ^. CO OS us o '• V ^ (N us V ^ rH OS t^ CD US us -r= us us US »(S Ph CO l-~ 00 OS US o o CO rH OS CI CM rH O O O OS o ph 1— < 1-1 1-1 rt r~i t-H rH i-H » ^1 rH US ITS US US us us n s t~- 1"-. 00 O 00 u> OS o 0O IM CO CO us <M <M rH O OS OS r-\ r— 1 >-^ ■— 1 rn rH rH rH [Zi K t~ r^ t- O OS OS ,— 1 1—1 • <N TX • ■>JI o TjH rX CSI O o OS T3 C 1 — l <M "■ us io "■ ~; "~ t^- CM o cm : T— I I— 1 M CO CO O 00 t^. CO i— I US US r« rH CO CO CO ou CM s " - rH rH O Ci OS OS CD us us us co : CO CO US US us US ^ ::' us us <D CO rH rH CO 00 CO OS : s 1 o c '" O O O) cow CO CD 1-1 1-1 c 1-1 O -# us • '.° CD CD t^ 00 O OS OS OS _ : o : i— i 5 o r-H OJ 00 »^ t^ r~ w. E 3 CO us te CO -* r* O CO J-~ t-. CO : os o : o o O O OS OS CO CO CO o T3 1 — 1 <M : CO : co : CO CD CO " r* r* rH CM (M US US o : os OS OS OS OS !>. CO CO CD •imojjo ■< >JS i— CCMCOrHUSCOWCOOSOrHCMCOrHUSCOWCOOSO CI £ E S E E fl s 3 3 3 3 E E s s E 3 3 3 3 g S § § § r=5 <N CO' ■"* US CO 262 PROFESSOR L. BECKER ON a faint band. The difference cannot be due to a change in sensitiveness of the photo- graphic film, as its effect is inappreciable within the extent of a band. On the other hand, it will be proved from the later plates that these bands must be similar also in intensity, and therefore other radiations must be superposed on the hydrogen radiations at the ends of the bands. Table IV. Table V. Mean of the Results from three Plates, Nos. 3, 4, and 5, showing the Similarity of Structure of the three H -bands and the more rapid Fallin«-ofF in Intensity of the HP-band. ,c2 Rs Hy H> [ShX a Inty. «o Inty. « Inty. 4 7 5 1 -43 -44 5 7 5 2 -32 5 -32 8 -33 7 3 -23 -24 -24 4* -21 7 - 22 9 -21 9 5 -18 9 9 6 \-u 8 -16 -16 5 2 7 8 -14 -14 10 11 8 -12 -12 11 -11 13 9 }-< 9 - 6 - 5 8 9 10 - 4 - 3 10 12 13 11 1 +1 + 2 + 1 min. 9 8 12 + 4 + 3 10 13 13 13 } +7 1 + 8 f + 8 = 10 7 14 ) + 10 10 13 13 15 \ + 13 1 } +13 min. fi } +12 = 9 16 1 J 10 + 14 12 J 12 17 + 17 + 18 + 16 10 9 11-5 18 + 21 8 + 20 11 + 21 9 8 19 + 25 + 25 + 26 -7 7 10 20 + 31 6 6 + 30 9-5 9-5 + 33 6 5 5 21 + 3G 6 + 38 9 5 Mean Structure of Bands. No. 1 2 3 4 5 6* 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 -73 -55 -43 -31-5 -23-9 -21-8 -17-5 -16-6 -14-4 - 12-0 - 5-8 - 3-8 + 1-2 + 3-2 + 6-8 + 9-5 + 12-4 + 13-0 + 16-8 + 20-1 + 25-1 + 31-7 + 34 + 36-4 + 40-5 + 56 Inty. 5 6 7 8 9 3 11 13 7 13 7 12-5 7 12-5 7 11-5 10 5-5 3-5 2-5 1 0-5 M ?'$ a S- CO 3 PQ Index Nos. of Plates. 3,4 2 I 5 2,3.4,5,6 3,4,5,6 1,3,4,6 3,4,5 The Fraunhofer lines occur only on Plates Nos. 3 and 4, March 18 to 20 and March 25. THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 263 6. The Continuous Spectrum. — The question whether the continuous spectrum has a share in these radiations is settled by photo-plate No. 1, which was orthochromatic. There are five minima recorded between wave-length 496 and 538, whose intensities ransre between and 0*3 degrees. I obtain the reduction for relative sensitiveness for different wave-lengths from a series of exposures on the sky spectrum. On the supposition that the intensity curves of the spectrum of the sky and of the star are identical, the intensity of the continuous spectrum of the star does not reach 1 degree within the area photographed, and probably is less, considering the observed colour of the star. Not to complicate the problem unnecessarily, I have disregarded the continuous spectrum, and it will be seen from what follows that the only effect this omission can have on the result is to increase slightly the calculated degrees of intensity of the bands. 7. Mean Structure of the Bands. — The continuous spectrum being very faint, the difference under discussion must be caused by superposed bands. From the plates taken after August I shall prove that not only the H-bands, but all the bands, are similar. I assume that the same holds good for the earlier plates, and that therefore the superposed bands are similar to the H-bands. The unknown structure of the bands thus enters not only the H-bands, but also the bands overlapping their ends, and that structure must be so determined as to give the observed degree of intensity. In the solution of this problem two other questions are involved : First, in what way does the intensity curve of the band vary if the intensity of its maximum alters ? Secondly, what is the resultant intensity on my degree scale if two radiations which singly produce certain degrees of intensity on the photograph be superposed ? The two bands, A = 4902 to 4959 and A = 4981 to 5046, agree in extent and in position of their maxima with the bands calculated from the bands of Table IV. by formula (l), and the wave-lengths of their zero, \, are found respectively to be 4922 and 5016. I compared the observed degrees of the H^-band, which in first approximation served as standard, with those at corresponding points of the faint bands A =4922 and 5016, discarding the points where the bands overlap, and deduced by interpolation the relations given in Table VI. For instance, a band which has at different points the Table VI. — Corresponding Degrees of Intensity. 13 11 9 7 5 3 1 11 9-3 7-5 5-8 4-1 2-3 0-6 9 7-5 6-1 4-6 3-1 1-7 0-2 7 5-8 4-6 3 4 2-2 1-0 5 4-1 3-1 2*2 1-3 0-4 3 2-3 1-7 1-0 0-4 1 0-6 0-2 264 PROFESSOR L. BECKER ON degrees of intensity 13, 11, 9, 7, etc., has at corresponding points the degrees, say, 5, 4-1, 3-1, 2-2, etc. (Also see § 9.) The bands contained on the first seven photographs here under discussion give no evidence as to the second question. On the later plates there are two bands which can with certainty be identified as consisting each of two bands, while the structure of the standard band is independently determined from detached bands. From Table XII., where the later observations are compiled along with the calculated bands, it will be seen that the sum of the calculated degrees of intensity due to radiations of the same wave-length nearly agrees with the observed intensity. I adopt this additive rule here as a working hypothesis, the accuracy of which will be investigated in § 9. I deduce the common structure of the bands by successive approximation. Choos- ing first the H^-band of Table IV., I calculate the band for A n = 4922, and employing Table VI., reduce the degrees of intensity so that the degree of intensity of the maximum of this band agrees with the observed intensity of the maximum. I then subtract the calculated degrees from their observed values, and find the degrees of intensity at the different points of the H^-band freed at its less refrangible end from the superposed band 4922. From the corrected Hp-band I calculate the band for X = 4265, which a preliminary discussion had shown to overlap the more refrangible end of the H y -band, and proceeding as before, I obtain the intensities at different points of the more re- frangible end of the H y -band. In second approximation I combine these results, unci repeating the calculation, find the mean structure as contained in Table V., the values a Q being the means, with regard to weights, of the measurements given in Table II. 8. Resolving of the Spectrum into Bands. — I set myself the problem to find the wave-lengths X n of the zero of each band, and the degree of intensity of its maximum, which I shall call the intensity of the band, so that the superposed bands represent the observed intensity curve. I found this research on the following basis : — 1. The continuous spectrum is faint and may be neglected. 2. The bands are similar to the band given in Table V., and determined by formula (l), A being unknown. 3. The intensity curve of each band is defined by the unknown maximum intensity and the data contained in Tables V. and VI. 4. At places where bands are superposed, the re- sultant degree of intensity is the sum of the degrees of intensity which the radiations would singly produce on the photographic plate. The last assumption is merely a convenient rule, which, though not strictly correct, is sufficient for our purpose, as will be proved in § 9. I may mention here the con- siderations which induced me to undertake a research which at first sight appears to be hopeless. I suppose that two bands have been identified in the spectrum, and draw their intensity curves as calculated from Tables V. and VI., together with the observed intensity curve of the spectrum. The length, in the direction of the axis of wave-lengths, of the area bordered by the three curves is independent of the manner in which the ordinates of the two bands are deducted from those of the observed intensity curve. THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 265 If this length agrees with the extent of one band, which is almost constant for the brighter bands, the position of a band can be adjusted in this area, i.e. the wave-length, X , of the zero of the band can be found. Fainter bands are shorter in length, and two or more fainter bands may be combined so that their total extent is that of a bright band. On the understanding that a band selected in the manner described may be replaced by two or more fainter bands whose values of \ differ little, the wave-length of the zero of the band is independent of the rule according to which the degrees of intensity are combined, and depends only on the wave-lengths A of the zeros of the two adjoining bands. The error of the rule will appear only in the residuals. To a lesser degree, this holds also good for a space belonging to two or more bands, and the greater the space to be covered, the more important it will be that the intensities be correctly compounded. For this reason I have divided the spectrum between H 3 and H y into four parts, introducing three bands, whose maxima I make to coincide with pronounced maxima of the observed intensity curve. The intermediate spaces were explained by as few bands as possible, with the intention that each could afterwards be replaced by two or more fainter bands, should this improve the agreement of the spectrum with one which otherwise resembled it. A. For the discussion of the spectrum between H^ and H v I employ the intensity curves of photo-plates Nos. 3 and 4. As before, I add 5 to the degrees of intensity of No. 4 to refer them to the same scale as was chosen for No. 3. The spectrum on No. 4 differs somewhat from those on the other plates, of which No. 3 is the best representative. On plate No. 4 there are four prominent maxima between H^ and H v , viz., 4459 to 4480, intensity 6 ; 4560 to 4589, intensity 5'5 ; 4627 to 4648, intensity 8 ; 4682 to 4691, intensity 8 ; all of them fading off several degrees towards both sides. I assume that they are due to bands whose \ is respectively 4470, 4574, 4637, 4687, and I calculate the intensity curves of these bands from the data given in Tables V. and VI. The calculated degrees of intensity were written out at intervals of two tenth-metres, and subtracted arithmetically from the observed intensities. To explain the residuals, I chose, in accordance with the above, as few bands as possible, and introduced further the condition that the same bands be selected for both photo-plates. Between 4341 and 4470 at least three bands were required for plate 3 and two for plate 4 ; between 4470 and 4574 three for plate 3 and two for plate 4 ; between 4574 and 4637 one each for plates 3 and 4 ; between 4637 and 4687 one for plate 3 and none for plate 4 ; between 4687 and 4861 two bands for both plates. By a lengthy process of trials in which the wave-lengths and the intensities of the bands were altered, including those of the above bands, I found the wave-lengths X of the zeros of the bands and the degrees of the intensity of their maxima, as given in Table VII. under the heading A. The three intensity curves calculated from these data at intervals of two tenth-metres are represented on Plates I. and II. under A, together with the observed curves, which are dotted. The straight lines drawn at the top of the plates show the extent of each band and the number of superpositions at each point. 26G PROFESSOR L. BECKER ON Table VII. further contains the hydrogen spectrum observed by Wilsing and the spectrum of Nova Aurigee by Vogel, both copied from the table in Wilsing's memoir Untersuchungen ueber das Spectrum der Nova Aurigce. I ought to mention here that during the progress of my work I did not consult any previous researches or observations on new stars, and that I arrived at the result A without bias. Between 4067 and 4341 the wave-length A of the zeros of the bands agree well with the wave- lengths measured in the spectrum of Nova Aurigse, and the two lines 4922 and 5016 are also present in both. On the other hand, there are marked differences between H^ and H y , the region which, owing to the large interval between two identifiable bands, presented the greatest difficulty to division into bands. The same remarks apply to the hydrogen spectrum. It is of no moment that some of the lines, as 4388, 4472, etc., are found intensely bright, because, in accordance with the above, each might be split into two or more fainter lines. The fact that the hydrogen line 4581 is the only bright line of intensity above 6 which is not represented in the spectrum of Nova Persei, while it occurs in the spectrum of Nova Aurigse, appears to suggest that the observed maximum 4560 to 4589 is not due to the band X = 4570, which is one of those used in subdividing the spectrum. B. I therefore repeated the work between H^ and H y , and subdivided the spectrum as before, but chose 4581 instead of 4570 ; 4634 instead of 4637 ; and 4684 instead of 4687. I further introduced the condition that as few lines as possible should be chosen, and that where the wave-length of the zero of a band fell near that of a hydrogen line, the wave-length of the H-line should be taken. The introduction of the line 4581 instead of 4570 as zero of a band changed the position of all the bands as far as the H r -band ; 4570 being the mean of 4559 and 4581, the two bands belonging to them share almost equally in producing the maximum formerly ascribed to 4570. Each band entails the introduction of a series of bands fitting into one another, and there are thus 14 bands required to represent the intensity curve compared with 7 bands before. In other regions I altered some of the wave-lengths slightly to make them agree with those of hydrogen. The band X = 4768, which does not occur in the hydrogen spectrum, is perhaps due to a series of faint lines. The result of this new analysis is given in Table VII., B, and the intensity curve calculated at intervals of 2 tenth-metres is drawn on Plates I. and II. under B. It is remarkable that the lines of the hydrogen spectrum, which I have been forced to take from Wilsing's table as being within 3 t.m. of the zeros of bands actually obtained, include all the hydrogen lines whose intensities exceed 2 between 4341 and 4861, though no heed was taken of their intensity. The spectrum as defined by A agrees well with that of Nova Aurigse. In the same table I have further entered all the lines of helium except those of the two second subordinate series. All of them have corresponding lines in the Nova spectrum. On the assumption of the additive rule, the brightest lines of the hydrogen and helium spectrum represent the intensity curve, each line being broadened according to Table V. and formula 1 ; and I consider the conjecture that hydrogen actually pro- THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 267" duced the spectrum not only possible, but probable, for these reasons: — (l) Towards- the more refrangible side of H Y , lines coinciding within admissible limits with prominent H-lines, were found without any bias, and they belong to that region of the spectrum which from the outset offered the least difficulty to the splitting up of the intensity curve into bands. (2) The intensity curve of the band belonging to 4637 or 4634, also detected without bias, follows the intensity curve of the March 25 plate in that part of the spectrum without requiring the introduction of another bright band, and this line is, together with Hp, H y , H 5 , the most prominent line of the hydrogen spectrum. (3) If, instead of the line 4570 found under A, which does not occur in the hydrogen spectrum, the prominent H-line 4581 is introduced, the lines then required to represent the intensity curve agree in position more closely than before with those of Nova Aurigse, i.e. a star whose spectrum is, as I shall show, identical with that of Nova Persei after August. (4) The lines of the hydrogen spectrum, which I have been forced to take from Wilsing's table as being within 3 t.m. of the zeros of bands actually obtained, include all the hydrogen lines whose intensities exceed 2 between 4341 and 4861, though no heed was taken of their intensities. 9. Proof that the Error of the Additive Rule does not affect the Result qf\ 8. — The results just derived rest on the assumption that the radiations of intensity i x . . . i n which individually give on the photo-plate a blackness of degree m x . . . m n for the same exposure t produce if acting together during the same time t a blackness equal to 2m. It is, how- ever, well known that this cannot be correct for all degrees of blackness. With the view of determining the error introduced by the use of the additive rule, I exposed several plates on a continuous spectrum, each plate containing five spectra, the exposures of which were proportional to 1, 2, 4, 8, 16. I estimated the degrees of blackness of the spectra in the same way as done on the star photographs. Any two degrees of blackness could then be superposed, and compared with those estimated for another exposure. I find that for the degrees of blackness occurring on the star photographs, the blackness of the film is about proportional to the time of exposure for a constant intensity of radia- tion, and that the degrees of my scale are about proportional to the blackness. Since this relation cannot hold good for the highest degrees of blackness, I take it to be only approximately true for the lowest degrees, and put (2) -1 = -Q. Lj for a radiation of constant intensity i, k fi?>h) and choose (3) /(m) = 10 om -l, where m is the degree of blackness, t the time, and a a constant. Scheiner's Die Photographie der Gestirne contains on p. 246 a table, the results of experiments by Michalke, which gives the times of exposure for intensities of radia- tion varying from 1 to 36, to produce the same degree of blackness on the photographic film. I find this table is sufficiently well represented by (4) ti b = constant 6 = l - 08, for a constant blackness m. TRANS. ROY. SOC. EDIN, VOL. XLI. PART II. (NO. 10). 41 268 PROFESSOR L. BECKER ON Table VII. — Spectrum of Nova Persei in March and April 1901. Nova Persei. Hydrogen H (Wilsing). Helium. Nova Auiigae * (Vogel). A H. Intensity Intensity r t o o CN O) V CO S id CN A . CO 03 1 CO o 1 CO ^1 A. ] A. I. Series. A. I. c3 a 3 a c8 a 1 o a ea a 4026 5 l 5 4026 5 l 5 4027 4044 4055 4 3 3 3 2 4026 5 11,1 ( 4063 6 6 4067 i 2 5 4067 2-5 • 4067 | 4070 4088 4097 5 5 3 2 5 5 1 1 4067 b 3 4102 9 10 10 4102 9 10 10 4101 5 1 4102 b 3 4122 2 2 2 41 32 3 3 3 4132 4 2 4125 2 4145 2 2 2 4145 1 1 4145 3 1 4144 2 II, 1 4160 1-5 1-5 1-5 f 4157 1 4163 4171 5 5 4 2 2 3 4158 2 4174 6 6 6 4177 55 4-5 4-5 4177 4182 4189 4196 8 3 2 3 8 1 3 4176 vb 3 4210 2 2 3 4208 1-5 1-5 1-5 j 4205 | 4213 4222 6 7 4 4 5 3 4234 4-5 4-5 4-5 4233 4-5 5 5 4233 4243 4253 3 1 2 1 1 1 4230 b 3 4265 3-5 4-5 45 4265 3-5 4-5 4-5 4265 4293 1 1 4262 4288 vb 3 b 3 4304 1 3 3 4305 1 3 3 4305 4312 4330 3 2 2 3 4315 b 3 4341 9 10-5 10-5 4341 4364 4377 9 2 1 10-5 3 10-5 2 2 4341 4377 4382 13 2 3 1 4341 4383 b 4 3 4388 8-5 8-5 ■ 8-5 4388 5-5 4-5 4 4389 4402 2 2 1 1 4388 3 II, 1 4410 2 4410 4 5 3 4410 4413 4419 4 2 2 3 4417 b 3 4436 5 5 3 5 J 4425 \ 4448 1 2-5 2 4425 4 4 4435 b 3 3-5 3 3 4448 2 4 4445 2 4459 ■ • • 3 3-5 4459 3 4 4472 4 8-5 9 4472 3 4 3 4472 6 1,1 4473 b 2 I. = intensity. I = helium, II = parhelium, p = principal series, 1= first subordinate series. In the last column b = broad, 3 = bright, 4 = very bright. * Copied from Publicutionen des Adrophysikalisclien Observatoviums zu Potsdam, xii. p. 96. THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 269 Table VII. — continued. Nova Persei. Hydrogen (Wilsing). Nova Aurigee (Vogel). A. B Helium. Intensity. Intensity. © o I>5 A . CO V CO 03 id p 00 1 — 1 -a o ,4 o t-i 00 *P A. ] [. A. I. Series. A. I. 3 p 3 3 3 3 3 4482 1 1 4480 ) 4488 3 3 4 4488 2 4 Us 4502 5 3 4500 3 4 1 J 4498 1 4501 2 2 4 4495 4507 1 2 4520 5 6 6 4522 2-5 3 5 1 4521 | 4523 2 1 2 3 4520 lb 3 4535 4 3 2 4535 4551 4555 2 2 2 2 2 2 4530 f 4558 4 25 2 4559 6 5 4 4559 5 2 4557 b 2 4570 7 8 8 4581 3 5-5 5 4581 4597 6 2 10 1 4583 b 3 4612 3-5 5 3 4608 2 4 1 4608 4619 2 2 1 1 4637 35 3-5 8 4634 5 6 9 4634 9 15 4628 b 3 4660 3 6 4662 4672 2-5 2-5 3 4662 4672 1 1 2 3 4687 5 5-5 10 4684 5 6 10 4684 4710 4719 3 2 2 3 2 2 4725 4-5 4-5 5 4724 4 4 6 4724 o 3 4768 5-5 55 5-5 4768 5-5 5-5 5-5 4797 1 4861 13 13 13 4861 13 13 13 4862 7 2 4862 b 3 4922 4 4 4 4922 4 4 4 4932 4973 2 4922 4 II, 1 4924 2 5016 3-5 0-5 2 5016 3 0-5 2 5014 5055 5016 6 II, p 5016 o 5132 1 5132 1 5178 2 5178 2 5167 vb 3 5200 2 5200 2 523 528 2 2 5327 2 5327 5405 3 2 5388 5317 4 5456 3 5451 3 5481 5500 2-5 5495 2-5 5499 5505 5544 4 5537 4 5537 5589 3 5584 3 5635 2-5 5640 3 5680 2 5689 2 5689 ' The hydrogen lines 4973 to 5689 were measured by Hasselberg. 270 PROFESSOR L. BECKER ON On p. 247 Prof. Scheiner further gives the results of his experiments on artificial stars produced by means of a Zoellner photometer. He finds that if the time of exposure be increased 2'5 times, the faintest stars recorded on the plate are only 0'7 mg. fainter than before, therefore b = 1*4. Let a radiation of intensity i produce a degree of blackness m, and a radiation of intensity i' a blackness m', both in time t. Let t! be the time required for i to produce m', then by (2) and (4) ti' b = t'i" and -, = $2?)-. therefore t f(m ) f(m) f i\'' (5) v / = -. for the same exposure on radiations of intensities i and i . ■ ' /(to') \i'J * I assume that the broadening of the lines into bands is due to the same physical cause, and that the ratio of the intensities of the radiations at any two corresponding points of two bands is a constant for these two bands ; therefore by (5) (6) ^" = constant, and ^7^/ = Jr~7\ = constant, /(to) fim') /(//) /(/,) - /(,*') where n and m are the degrees of blackness at two points of a band, and 11! and m' those at corresponding points of a second band. By means of (6) I determine the constant a in f(m) from the observed corresponding degrees of blackness contained in Table VI. The result is a = 0'04, with which I have calculated the following table. Table VIII. Calculated corresponding degrees of blackness for a = O04. H- 13 11 9 7 5 3 1 11 9-2 7-4 5-7 40 2-3 07 9 7-4 5-9 4-5 3-1 1-8 0-5 7 B-7 4-5 34 2-2 1-2 0-4 m - 5 4-0 31 2-2 1-5 0-8 0-2 3 2-3 1-8 1-2 0-8 0-4 o-i 1 0-7 0-5 0-4 0-2 o-i For instance, if the maximum of a band of degree 13 is reduced to degree 5 in another band, blacknesses 7 and 5 at other points of the band become respectively 2 '2 and 1*5, while Table VI. gives 2 '2 and 1*3. The quantities in this table differ from those in Table VI. for all degrees of blackness greater than 0*8 by less than 0"2 degrees, and the average error is 0"1, but the differences increase to 0*4 for the degrees lower than 0'8. The function (3) therefore represents the observations satisfactorily. For a = 0*03 and 0*05 the residuals are respectively 30 and 50 per cent, higher than for a = 0'04, and for THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 271 <j = - 07 they increase to 2^ times the amount for a = 0'04. For double the exposure the degrees of blackness 1, 3, 5 and 10 become respectively 1*9, 5'3, 8*4 and 15. Let the intensities of radiation i x . . . i n give respectively the blacknesses of degree <m x . . . m n when exposed time t. The question is, what is the degree m if a radiation of intensity 2{ be exposed the same time t ? Formula (5) gives i B F(m ) = F(ra„)2i where F(ra) = (/(m))» , therefore (7) F(m ) = 2F(ro M ). An extract of the F-Table is given below. Table IX. m. F(m). m. F( n). 6=1-08. 6=1-4. 6=1-08. 6 = 1-4. 1 2 3 4 5 6 7 00 o-ii 0-23 0-35 0-47 0-61 0-75 0-91 o-oo 0-19 0-32 0-44 0-56 0-68 0-81 0-93 8 9 10 11 12 13 14 1-08 1-27 1-47 1-68 1-92 2-17 2-45 1-06 1-20 1-34 1-49 1-65 1-82 2-00 For instance, radiations which would singly give the degrees 1, 2, 3, 4 produce when superposed m =8'4 for 6 = 1-08 (by Table IX., F(m ) = 011 +0'23 + 0*35 + 017 = 116) and 11*1 for b = 1'4, as compared with 2m =10. The difference ?n — 2m depends on 2m and the number of radiations which are superposed. It varies most for the intensity curve which shows the greatest range between maxima and minima, and which has a different number of superposed bands at different places. On this account the calculated intensity curve, 2m, belonging to photo- plate No. 4, which on hypothesis B has from 3 to 6 superposed radiations in the region between H^ and H y is the most likely to differ from an intensity curve m calculated by formula (7). I assume the same bands, and in first approximation the same degrees m which contribute to the 2m-curve of Plate I., but I compound them according to formula (7). The resulting ?w -curve is then brought to agreement with the observed intensity curve by suitable changes of the intensities of the bands, and the question is whether this curve satisfies the observed curve as well as the 2m -curve does. Table X. contains the calculation in detail at an interval of 10 t.m., while it was made for every 5 t.m. The first columns show, under the heading m, the degrees of blackness at each point which the radiations would singly produce on the film. 2m is the ordinate of the calculated intensity curve on Plate I., which was made to agree as near as possible with the observed intensity curve (see § 8) ; m is the resultant if the degrees m be compounded 272 PROFESSOR L. BECKER ON according to formula (7). m -2m ranges between — 0*6 and —2*2 for 6 = 1*08, and between +0*4 and + 2*4 for 6 = 1*4. Let Smx . . . Sm n be the corrections of the degrees m x . . . m n which change the degree of blackness m by Sm therefore, (8) S^Ogm,, = dF ( m °hm . dm n dm A change Sm-^ at wave-length X of band 1 can be brought about only by all the degrees of blackness being changed at every place of this band. I express Sm 1 by the change Snj of the degree of blackness /*i of the maximum of band 1 ; m x being the quantity which was shortly called the intensity of the band, and I do the same for all the indices 1 to n. Employing (6) I replace in (8) $m n by <V» and eliminate function /by F. (9) 2F(m/ lo g/fc"V = dF }™°hm . dfi n dm^ To reduce the work of calculating, I change the degrees of the maximum of every band by an amount <V n determined by (io) g= di °g/foy, dfx n and obtain from (9) and (7) (11) ^logFK), dm (m ) = m + 8m . I determine x by (ll) with Sm = ^m — m , and calculate $m at each point from the mean value of x. For 6=1'4, //. of band \ = 4425 was reduced by 1 in addition to <V, and the intensities of the last four bands were not changed at all. (m ) = m + $m which appears in Table X. is then the calculated intensity curve if the degrees of blackness of the maxima of the bands given in Table VII. be changed by certain amounts to be calculated from (10), and the degrees at each point be compounded ac- cording to (7). The last three columns give the residuals left in the observed intensity curve. 2m differs from the observed intensity curve on an average 0*58 degree, (m ) differs 0'57 for b= T08, and 0'63 for 6= 1*4, while (m ) differs on an average 0*26 from 2m. The observed intensity curve is therefore equally well represented by 2m and by (m ), and therefore the hydrogen lines of Table VII. represent the spectrum, no matter whether the degrees of intensity of their bands be compounded by mere addition, or according to formula (7). Combinations of bands which, if compounded according to the additive rule, leave inadmissible residuals in the observed intensity curve, must give errors of the same order if formula (7) be employed ; and I conclude that if this formula had been used at the outset in analysing the spectrum into bands, the result would have been identical with that contained in Table VII. A similar calculation for the intensity curve of photo-plate No. 3, assumption A, gave an average error, observed m — 2m = 0*63, observed m — (m ) = 0'59 for 6=1*08, and 0'58 for 6 = 1*4, while (m ) differs on an average 0'25 from 2m for both values of 6. This agreement proves again that the use of the additive rule cannot have influenced the analysing of the spectrum into bands. THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 273 Table X. Calculated Intensity Curves, 2m and (m ),for Photographic Plate No. 4. O M T2 4360 70 80 90 4400 10 20 30 40 50 60 70 80 90 4500 10 20 30 40 50 60 70 80 90 4600 10 20 30 40 50 60 70 J 80 90 4700 12 9-5 9 8-2 7 7 7 7-5 10 11 10 10 8 9 9 9-5 9-5 8-5 8 10-5 10-5 11 13 13 12 10-5 11 11 13 12 Intensity Curves of the different Superposed Bands. 6-5 4-2 0-2 0-4 0-6 ro 1-4 3-0 3-0 2-8 1-1 06 08 1-1 1-6 21 4-0 4-0 3-8 21 ro 0-4 2-4 3-6 4-4 4-8 6-0 8-4 10 9 6 2-0 1-9 12 0-4 0'9 2-0 1-9 1-6 0-4 o-i 1-0 1-0 0-9 0-2 0-2 0-7 2-1 1-9 1-6 05 0-4 0-5 0-6 l'l 1-6 22 4-0 3-8 3-6 2-1 1-0 0-2 1-0 0-9 0-8 0-8 1-7 2-2 2-5 3-4 1-6 2-0 4-0 3-8 3-8 1-9 0-4 0-8 1"2 2-3 3-0 2-8 IT. 0'5 0-4 1-3 17 2 2 2-7 5-0 5-0 4-8 27 1-5 0-4 0-6 1 1-4 3-0 3 2-8 1 5 0-5 0-4 1-3 1-7 2-2 2-7 4-1 5-0 4-8 3-8 1-5 0-6 8 1-3 1-7 3-5 3-5 3-3 1-5 0-1 0'4 0-9 2-0 T9 1-6 0-4 11-2 10-8 8-1 7-0 8i 7-0 7-0 7-0 8-2 9-2 10-6 10-9 10-1 8-5 8-5 8-3 8-6 9'7 9-5 9-:: 7-1 10-7 10-3 9-9 7-6 8-1 107 12-6 13-0 12-1 10-3 11-4 12-2 12-1 12-4 by Formulas (7) and (11). 6 = T0?. 9-4 8-9 7-2 6-4 7-2 6-2 6-3 6-2 7-0 7-8 8-7 8-9 8-4 7-4 7 3 7-2 7-6 8-4 8-0 7-9 6-5 9-0 8-8 8-5 6-8 7-2 95 11-0 11-1 10-4 9-2 10-1 11-2 11-0 10-9 i (m ). 6=1-4. + 1-8 10-8 + 1-9 10-2 + 0-9 83 + 0-6 7-4 + 0-9 8-3 + 0-8 7-2 + 07 7-3 + 0-8 7-2 + 1-2 8-1 + 1-4 9-0 + 1-9 10-0 + 2-0 10-2 + 1-7 9-7 + 1-1 8-6 + 1-2 8-3 + 1-1 8-3 + 1-0 8-8 + 1-3 9-7 + T5 9-2 + 1-4 9-1 + 0-6 7-5 + 1-7 10-3 + 1-5 10-1 + 1-4 9-8 + 0-8 7-9 + 0-9 8-3 + 1-2 10-8 + 1-6 12-5 + 1-9 12-6 + 1-7 11-9 + 1-1 10-5 + 1-3 11-5 + 1-0 12-7 + M 12-3 + 1-5 124 12-2 12-2 9-6 8-4 9-6 8-8 9-0 9-4 10-4 10-8 12-0 12-2 11-4 10-0 10-4 10-0 10-4 11-0 10-8 10-6 8-1 11-2 11-0 10-8 8-6 9-4 11-2 12-4 12-6 11-8 11-0 11-8 12-2 12-0 12-4 W -1-0 -1-4 -1-5 -1-4 -1-5 -1-8 -2-0 -2-4 -2-2 -1-6 -1-4 -1-3 -1-3 -1-5 -1-9 -1-7 -1-8 -1-3 -1-3 -1-3 -1-0 -0-5 -0-7 -0-9 -1-0 -1-3 -0-5 + 0-2 + 0-4 + 0-3 -0-7 -0-4 o-o + 0-1 o-o (m ). 107 107 8-3 7-2 8-0 7-0 6-7 7-1 8-2 9-1 10-5 107 9-9 8-6 9-0 8-6 9-0 9-6 9-4 9-2 6-9 10-0 10-0 10-3 8-1 9-1 11-2 12-4 12-6 11-8 11-0 11-8 12-2 12-0 12-4 Kesiduals I O + 0-8 -1-3 + 0-9 + 1-2 -1-1 o-o o-o + 0-5 -0-2 -1-2 -0-6 + 0-1 -o-i + 1-5 -0-5 + 0-7 + 0-4 -0-2 o-o -0-8 + 0-9 -0-2 + 0-2 -1-9 + 0-4 -0-1 + 0-3 + 0-4 o-o -0-1 + 0-2 -0-4 -1-2 + 0-9 -0-4 - (m ). + 1-2 -0-7 + 0-7 + 0-8 -1-3 -0-2 -0-3 + 0-3 -o-i -1-0 o-o + 0-8 + 0-3 + 1-4 -0-3 + 0-7 + 0-2 -0-2 + 0-3 -0-6 + 0-5 + 0-2 + 0-4 -1-8 + 0-1 -0-3 + 0-2 + 0-5 + 0-4 + 0-1 o-o -0-5 -1-7 + 0-7 -0-4 + 1-3 -1-2 + 0-7 I + 1-0 -1-0 o-o + 0-3 + 0-4 -0-2 -1-1 -0-5 + 0-3 + 0-1 + 1-4 -1-0 + 0-4 o-o -o-i + 0-1 -0-7 + 1-1 + 0-5 4-0-5 -2-3 -o-i -1-1 - 0-2 + 0-6 + 0-4 + 0'2 -0-5 -0-8 -1-2 + 1-0 -0-4 274 PROFESSOR L. BECKER ON The Spectrum from 1901 August 1 to 1902 November. 10. The Mean Spectrum. — The results derived from the photo-plates Nos. 8 to 21,. 1901 August 1 to 1902 January, agree closely with each other, and it is unnecessary that the results be given separately. Those derived from the later plates, Nos. 22 to 27, are considerably less accurate, owing to the faintness of the spectrum, but they suffice to show that the bands did not change in position. Their results are also not printed separately. The changes which the spectrum underwent belong to the intensity, and they appear for the whole period in Table XVII. I combined the wave-lengths and the estimates of intensity to mean values, which are given in the first columns of Table XII. For most bands seven to eight plates contributed to the mean, and for the band near H y thirteen plates, all belonging to the period August to November. The two bands at wave-lengths 386 and 397 were outside the range of the plate until the beginning of October, and they rest on the results of the plates Nos. 18, 19, 21. In Table XII. the intensities of the bands, therefore, do not belong to the same epoch. The average error of a tabulated wave-length is 0*3 t.m. With reference to the faint bands, whose intensities do not exceed 1*5, and which were difficult to measure, most of the detail had to be discarded, because it was seen only on one plate. The neglected measurements are about five per cent, of the total number. The wave-lengths of these faint bands may be several tenth-metres wrong. 11. The Common Structure of the Bands. — Of the detached bands, the first two have the most pronounced intensity curve (see Plate III.). I shall show (see § 12) that their structure is similar, and further, that the similarity extends to all the other bands. The wave-lengths of the lines to which the bands belong being unknown, I introduce \ n , the mean of the wave-lengths of the three minima, and determine a m from (12) ^.. + «-iW0' a m is given in Table XIII. for the first two bands, and two other prominent bands, whose zeros X TO equal 4364 and 4726. The latter merge into fainter bands, which overlap their more refrangible ends ; the intensities at these places are bracketed in the table, and I do not take them into account here. The adopted values of Table XIV. are the means of the figures contained in Table XIII. , with the exception of a few which were corrected so as to represent other bands better. I have also calculated a m for all the other bands except one, and drawn on Plate III. their intensity curves with a m as abscissa. 12. The Calculated Spectrum. — I decide whether all the brighter bands are of the same type by calculating the different points of the bands from a m of Table XIV. by means of formula (12). About six well-defined points contributed to the final X m , and its calculated error is on an average 0"2 t.m. The degrees of intensity have been obtained from Tables VIII. and XIV. I have given the calculated bands in Table XII. Including all points, I find that the calculated wave-lengths differ from the observed wave-lengths on an average 0'5 t.m., as compared with a calculated average error of 0'3 t.m. for A, of THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 275 Table XI- -Photog raphic j Plates, 1901 August to 1903 Februai T <D Number of +3 c3 Points Measured S 5 Z3 60 r< Definition o a<X A c3 '~ H en « 53 Date. «| Remarks. C4H 4-3 M J> H O a | Star. Fe, Ti. S <D P -Q 3 fe Hours. Degrees Centigrade Star. Fe. B s 8 1901. Aug. 1 1-7 0-5 4 3 Intensified 35 25 4 6 9 Aug. 1 5 4-0 2-0 2 2 Intensified 93 24 15 8 10 Aug. 21 6-5 2-9 2 2 98 32 14 6 11 Aug. 26 5-8 1-8 2 2 109 40 18 4 12 Aug. 27 59 1-5 3 3 Intensified 77 24 10 3 13 Aug. 30 4 3 1-8 2 2 Intensified 94 25 14 4 14 Sept. 4 7-7 1-9 2 2 Very clear 114 33 20 4 15 Sept. 20 4-7 0-2 4F 3 Intensified 49 17 5 2 16 Oct. 2 Oct. 4 3-7 1-9 3-0 4F 3 Intensified 45 25 2 2 17 Oct. 6 7-0 1-3 2 2 Intensified 108 40 15 4 18 Oct. 31 8-5 1-2 2 2 64 30 12 4 19 Nov. 1 6-0 0-2 2 2 77 36 12 4 20 1901. Nov. 13 5-5 0-5 2 2 Intensified 133 45 17 4 21 1902. Jan. 12 5-8 11 2 1 Intensified 87 50 5 4 22 Jan. 26 Jan. 28 1-6 6-3 0-7 F 2 Intensified 77 45 2 4 23 Jan. 29 Jan. 31 7-6 7-6 2-2 F 1 Intensified 80 35 1 4 24 Feb. 9 Feb. 11 Feb. 12 Mar. 21 4-8 4-8 4-3 2-0 2-0 F 2 Intensified 81 42 4 4 25 April 1 April 2 April 17 April 26 April 30 May 1 May 2 2-6 1-8 1-5 1-5 0-8 0-9 1-4 5-0 vF 1 Intensified 78 52 2 4 26 Oct. 20 Oct. 26 Oct. 30 Nov. 1 2-2 9-0 33 8-5 33 F 2 Intensified 98 48 3 4 27 Nov. 17 Nov. 18 Nov. 20 1902. Nov. 21 1903. Jan. 7 Jan. 24 Feb. 1 6-7 6-7 6-9 2-2 0-5 0-5 0-5 1-7 vF 1 Intensified 46 32 2 1 4 "Width of slit:— 0-018 mm. for plates Nos. 8 to 25; 0-020 mm. for plate No. 26; and 0-022 mm. for No. 27. Angle of inclination of plate:- 13° for Nos. 8 to 17 ; 16° for Nos. 20, 22, 23, 25; 30° for Nos. 18, 19, 21, 24, 26; 8° for No. 27. Definition:— 1 excellent, 4 inferior, F faint. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 10). 42 276 PROFESSOR L. BECKER ON Table XII. The Observed Spectrum, 1901 August 1 to 1902 January, and the Calculated Bands. Observed. Calculated. Observ ed. Calculated. No. of No. of A Intensity. point Table XIV. A Intensity. A Intensity. point Table XIV. A Intensity. 3813 0-5 A ,,, = 3869-5 A M = 3968-0 3835 3853-5 3856-0 3 1 o 53-6 56-3 2-1 3978-5 3980-7 6 o 14 1 15/ 16 78-0 80-3 6-2 3858-1 4 3 58-1 3-2 3982-3 £1 1 17 81-9 2-6 7 9-5 3986 0-9 3859-4 11 4 59-6 11-5 4027 18 85-8 3861-1 8 5 61-4 9-5 4045 1 3862-6 1 6 62-7 1-1 4063 3863-8 3 7 63-9 2-1 4071-5 0-5 3865-2 11 8 65-2 10-5 4079-7 1 3867-9 9 68-3 o Kn > 6 5-3 4082-6 3870-8 10 70-5 10 10-5 2 1 861 3872-9 11 72-8 0-4 9 95 4088-3 2 89-4 3874-1 12 74-1 1-5 0-7 3876-5 5 13 76-4 3-2 4091-1 3 91-3 2-6 3878-9 10 8 14) 15/ 79-2 8-4 7-4 3 4 5 92-9 94-8 3-4 3881-4 16 81-5 2-6 3883 3 17 83-1 3-2 4095-8 6 96-2 0-2 1 11 1 7 97-5 3889 18 86-9 0-4 3900 0-3 4098-1 2-5 8 98-8 3 3936-6 > .„, = 3968-( ) 4102-8 0-7 9 02-1 1-2 3943-2 0-5 4105-3 10 04-5 3-0 3951-81 1 1 51-3 2-5 11 06-9 2 6 0? 1-7 4109-2 12 08-3 3954-3 2 54-4 0-5 0-7 3955-7 2 3 56-3 2-6 4110-4 13 14 ( 15 j 10-7 2-2 3957-5 4 4 57-9 8-0 2-5 13-7 19 3959-4 10 5 59-7 9-9 4115-3 16 16-2 0'7 3961-4 7 6 61-1 8-0 1-5 17 17-8 0-2 0-9 4121-3 18 21-8 2 7 62-2 3963-9 8 63-6 1-7 4140 ± - 5 3966-5 10 9 668 9-0 4165 ± o 3969-3 4 10 69-0 4-4 4220 ± 0-5 9 11 71-3 9-0 4265 ± 3972-9 12 72-8 8-0 4300 ± 0-3 3974-4 3 13 75-0 2-6 4306 0-7 7 7-1 4314-2 1 THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 277 Table XII. — continued. Observed. Calculated. No. of No. of No. of A Intensity. Point Table XIV. \ Intensity. Point Table XIV. \ Intensity. Point Table XIV A Intensity. X „ = 4341-0 43232 1-7 1 22-8 0-6 4326-6 2 2 26-1 10 4328-8 4 3 4 5 28-2 29-D 31-9 3-6 4-5 3-6 4333-2 1-5 6 7 33-5 34-7 0-3 0-6 4335-9 4 8 36-2 4-0 4340-1 9 39-6 1-7 4342-0 4-5 10 11 42-2 44-7 4-0 3-6 \ m = 4364-5 4346-3 1 12 46-2 1-0 1 46-2 2-0 4347-4 6 13 48-7 4349-6 9 14 ) 3-0 2 49-6 30 43517 15/ 51-9 3 51-6 9-0 4 53 3 12 16 54-5 2-6 11-0 4355-4 7 17 56-2 1-0 5 55-4 9-0 4357-4 1 0-3 6 7 56-9 58-2 10 2-0 4359-0 10 8 59-6 10-0 4361-4 9 18 60-5 4363-3 9 63-1 5 5-0 4365-3 10 10 65-7 10-0 4367-9 8 11 68-2 9-0 4369-8 3 12 69-7 3-0 4371-8 8 5 13 72-3 8 5 4374-5 14 74-9 4375-7 15 76-1 7 7 4378-4 3 16 17 78-1 79-8 3 1 4382-0 1 ■ 18 84-1 27S PROFESSOR L. BECKER ON Table XII. — continued. Observed. Calculated. No. of No. of No. of A Intensity. Point Table XIV A Intensity. Point Table XIV A Intensity. Point Table XIV A Intensity. 4388-7 0-8 43933 0-2 4398 4405 0-3 44467 1 4457-4 1-5 4488 4503 1-5 4554 4570 i 1-5 1 4578 3 K=4 1612-6 4590 1 93-2 1-5 2 96-8 0-2 0-3 4598-8 3 99-0 1-3 4 00-8 2 5 03-0 1-7 1-3 4604-4 1-5 6 7 8 04-6 05-9 07-5 01 0-2 1-5 4611-0 9 11-1 (> .,,, = 4635-3) 1 0-6 4614-5 10 139 1-5 1-5 11 16-5 1-3 1 (15-8) 4619-3 , 12 1 13 18-1 0-3 K= 4642-0 0-8 20-8 2 (19-4) 1-2 2 3 (21-6) 4-3 4623-3 14, 15 J 1-1 1 22-5 4 (23-4) 24-2 0-8 56 2-5 16 26-9 0-9 2 261 1-2 5 (25-6) 4-3 4627-7 0-3 3 28-3 6 (27-3) 0-4 17 28-8 4-3 7 (28-6) 0-8 5 01 4 30-1 5-6 8 (30-1) 18 33-3 5 32*3 4-3 5-0 4634-0 6 34-0 9 (33-9) 2 0-4 2-2 THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 279 Table XII. — continued. Observed. Calculated. , No. of No. of No. of A Intensity . Point Table XIV A Intensity Point Table XIV A Intensity Point Table XIV A Intensity. 7 35-3 0-8 4636-4 5 1 8 36-8 5-0 10 11 (36-5) (39-2) 5-0 4-3 4641-1 3 9 40-6 2-2 12 (40-9) 1"2 4643-5 5 10 43-2 5-0 ! 13 (43-6) 4645-3 14 , 15 J 3-8 11 45-9 (47-0) ,4-5 4-3 33 I 19 1 13 47-6 4648-9 50-3 1-2 16 (49-7) 4 ,14 I 15 3-8 1-2 4652-0 53-7 17 (51-6) 3-5 3-3 0-4 4655-0 3 16 56-4 1-2 18 (56-1) 4658-4 1 X m = 4687-8 17 18 58-3 62-8 0-4 4669-1 3 1 2 68-1 71-8 0-9 1-4 4673-6 3 73-9 5 4 5 75-8 78-0 4-8 6-2 4-8 4678-8 1 6 7 79-7 81-0 5 0-9 4681-1 6 8 \ 9 I 10 82-6 86-3 55 4687 2-5 89-1 5 5-5 4691-4 11 91-8 4694-2 4 2 12 93-4 4-8 1-4 1 ,,= 4715-6 95-8 4697-2 5 13 96-1 4-2 0-9 4699-7 7 14, 15 i 16 99-6 02-4 3-6 1-4 2 3 4 99-5 01-7 03-5 1-4 4-8 4704-0 6 17 04-3 0-5 5 05-7 6-2 4-8 K ,= 4726-1 4707-4 2-5 18 08-8 _ 6 7 07-4 08-8 0-5 0-9 l 06-3 1-4 280 PROFESSOR L. BECKER ON Table XII. — continued. Observed. Calculated. No. of No. of No. of A. Intensity. Point Table XIV. A Intensity. Point Table XIV. A Intensity. Point Table XIV. A Intensity. A m = 4715-6 A ,, = 47261 4710-2 9 8 10-4 5-5 2 09-9 2-2 4712-5 11 9 10 14-1 16-9 2-5 5-5 3 4 5 12-1 14-0 16-2 7-1 8-8 7-1 4717-2 7 11 19-6 4-8 6 7 17-9 19-3 0-7 1-4 4721-0 10 12 21-3 1-4 8 20-9 8-0 4725-0 7 13 "1 15 J 24-0 4-2 9 24-6 3-8 4727-0 9 27-4 3 6 10 27-4 8-0 16 30-3 11 30-1 1-4 7-1 4731-7 2 17 32-2 0-5 12 31-8 2-2 4735-4 18 36-8 13 15 i 34-5 6-2 5-5 38-0 5-4 4740-3 3 16 40-8 2-2 4742-6 1 17 42-7 0-7 4747-3 0-5 18 47-3 4757 0-3 4768 4776 0-5 4786 4799 0-5 4810 4824 0-5 4834-2 4840-3 1 A m = 4862-8 4842-5 1 1 42-4 0-7 4845-4 2-7 2 46-2 1-1 4848-3 5 3 4 48-4 50-4 4-0 5-0 4851-9 4-5 5 52-6 4-0 4853-8 1 6 7 54-4 55-8 0-4 0-7 4856-9 4 8 57-4 4-5 4861-7 1-5 9 61-3 2-0 THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 281 Table XII. — continued. Observed. Calculated. Observed. 1 Calculated. No. of No. of A. Intensity. Point Table XIV. A. Intensity. \ Intensity. Point Table XIV. A. Intensity. 4864-2 10 64-1 A m = 4959-4 4-0 4-5 14 , 15 J 4866-6 3-5 11 66-9 4-0 71-9 I 13 68-6 1-2 4870-0 71-4 1-1 4973 0-8 16 74-8 0-4 2-8 3-4 4978 17 768 4875-4 } 15 J 75-0 18 81-7 0-1 1-5 2-9 0-2 4878-7 16 77-9 1 11 K 17 79-9 4883-1 18 84-6 0-4 4986-2 0-5 1 86-2 0-7 4886 9 90-1 11 4900 0-5 0-3 49921 3 92-4 4-0 4917 4924 ; l m = 4959- 1 5 4 5 94-4 96-7 5 4-0 4938 0-5 1 l 2 38-6 42-4 0-3 4998-3 0-5 6 7 98-6 oo-o 0-4 0-7 4944-8 3 44-7 0-4 5001-0 4 8 01-6 4-5 4 46-7 17 5006-5 9 05-6 2-0 2-5 5 490 2-3 5009-9 10 08-5 4-5 4950-5 6 50-8 1-7 3 11 11-4 4-0 0-4 7 52-3 o-i 5017-5 1 13 13-2 11 4954-2 8 53-9 0-3 16-1 2 9 10 57-9 60-7 2-0 0-8 2-8 15 J 19-8 3-4 2-7 4964-0 11 63-6 2-0 5022-0 1-7 16 22-8 1-1 1 12 65-4 1-7 5026-1 0-7 17 24-8 0-4 13 68-2 0-4 5031 0-3 18 29-7 1-5 5053 282 PROFESSOR L. BECKER ON Table XIII. — Table of a m of Corresponding Points of Four Bands and of the Intensities, showing that the Bands are similar in regard to their Wave- Lengths after August 1, 1901. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 a in Tenth-metres. 3869-4 -18-5 -15-6 -131 -11-6 - 9-6 - 7-9 - 6-5 - 4-9 - 1-7 1-6 4-1 5-5 8-3 3968-0 4364-4 4726-3 -15-5 -13-9 -11-9 - 9-8 - 7-5 - 4-6 - 1 + 11-0 + 14-0 + 15-8 + 22-8 + 1 + 11-9 + 14-4 + 16-2 + 20 -15-3 -13-1 - 9-3 - 7-2 5-6 1-1 0-9 + 10 + 11 + 14 -15-3 -13-1 + 13 + 15' + 18-1 I +20-0 Degrees of Intensity. 3869-4 3 4 7 11 8 1 3 11 6 10 9 5 10 3968-0 2 4 10 7 2 2 10 4 9 9 3 4364-4 ( 9) (12) (12) 7 1 1 10 5 10 5 7 3 (3) 4726-3 ( 9) (11) (ID (11) ( 7) ( 7) 10 7 9 9 2 5-5 5-5 3 1 Table XIV.— Structure of the Bands after August 1901. Mean Adopted dm Degrees of T. M. Intensity. -18-9 -15-4 3 9 11 9 - 13-3 - 11-5 - 9-4 - 7-8 - 6-5 J 2 - 5-0 - 1-4 10 5 10 9 3 8 5 + 1-2 + 3-8 + 5-4 + 8-0 + 10-7 + 11-9 + 14-0 i 3 1 + 15-8 + 20-2 0*2 t.m. for a m , and 0*2 t.m. for A. TO . The observed degrees of intensity also agree satisfactorily with the calculated ones, or their sums at those places where two or three bands are superposed. The average difference, apart from signs, is 0'7 degrees of intensity; 91 of the discrepancies lie between and 0"5, 60 between 0*5 and 1, 26 between 1, and 2 and 6 are greater. I consider it therefore proved that the bands are in every way defined by \ n , the degree of intensity of their maximum, and the quantities given in Table XIV. 13. Permanency of Structure. — Table XV. gives the number of observations of a minimum, and the period during which it was observed. The bands A m = 3869 and 3968 were outside the range of the photo-plates Nos. 8 to 17. Their position is never- theless well determined, since the plates Nos. 18 and 19 contain all the six minima, and Nos. 20 and 21 each four. The first minimum appears to have been the most pro- nounced. In all, it was recognised 90 times, against 47 for the second minimum and 38 for the third minimum. Of the total of 177 minima, 158 belong to the 1901 plates. The number of the minima that have been found seems to depend on the brightness of the bands, and still more on the linear width of the minimum, which at wave-length 3870 was 0"23 mm., and at 5006 only 0'08 mm. In conformity with this, the THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 283 majority of the minima detected on the plates of 1902 belongs to the first two bands, which at first were the brightest of the spectrum, and then second in brightness only to the 5006-band. They are present on each of the three photo-plates on which they were in focus, plates Nos. 21, 24, 26, 1902 January to November. During this time there is also no change in the extent and the position of the maximum of the bands. The evidence therefore points to the conclusion that from August 1 1901 to the end of 1902 the structure of the bands remained unchanged. Table XV. — Table showing the Number of Photo- Plates on which the Minima have been measured, and the Periods to which they belong. Km Minimum 1. Minimum 2. Minimum 3. of Bands. 5 1 Period. <3 Period. CD 6 a Sz< 4 Period. 3869-5 6 1901 Oct. to 1902 Mar. 4 1901 Oct. to 1902, Mar. 1901 Oct. to 1902 Nov. 3968-0 4 Oct. to Jan. 6 Oct. to Nov. 4 Oct. to Jan. 4103-4 7 Aug. to Nov. 6 Aug. to 1903 Jan. 5 Aug. to May 4341-0 8 Aug. to 1901 Nov. 7 Aug. to 1902 Nov. 5 Aug. to 1901 Nov. 4364-5 13 Ausj. to 1902 Jan. 9 Aug. to 1901 Nov. 7 Aug. to Nov. 4612-6 1 Sept. 4. 3 Aug. to Nov. 1 Aug. 15. 4642-0 4 Aug. to 1901 Sept. 1 Aug. 26. 4687-8 9 Aug. to Nov. 8 Aug. to Nov. 4715-6 11 Aug. to Nov. 4726-1 8 Aug. to Nov. 5 Aug. to Nov. 3 Oct. to 1902 Mar. 4862-8 7 Aug. to Oct. 2 Aug. 21, Sept. 4. 4959-4 4 Aug. 1 Nov. 13. 5007-2 8 Aug. to 1903 Jan. 3 Aug. to 1901 Sept. 1 Sept. 4. A fourth minimum was measured twice in band A.,,, = 4364.5 on 1901, August 21 and 26. 14. Identification of the Bands. — The wave-Jengths A m of the zeros of the principal bands are compiled in the first column of Table XVI. Five of these can with certainty be identified as the hydrogen lines H^, H v , H s , and the two principal nebular lines. A m is the wave-length of an arbitrary zero of the band, viz., approximately the mean of the wave-lengths of the three minima. I change the zero and make it coincide with the wave-length \ of the line to which the band would be reduced under ordinary conditions. Let [s] be the mean of the corrections for the orbital motion of the earth on the days on which the photographs were taken, then A , . r.-i . A X = Ao + %5()0 = X "' + W + rt '"4500 (13) 8a m = a n 4500,, r , . N •a m = - T — (Am + LsJ -A ) K = K 4500 8a m + [s] TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 10). 43 284 PROFESSOR L. BECKER ON Table XVI. Sa„ Am W \ Hydrogen and Nebulae. 8om 3869-5 + 0-1 3968-0 0-0 4103-4 + 0-3 4101-8 + 2-1 4341-0 + 0-3 4340-7 + 0-6 4364-5 + 0-3 4612-6 + 0-4 4642-0 + 0-4 or (4635-3) 4687-8 + 0-4 + 0-4 (4633-8) 4715-6 + 0-4 4726-1 + 0-4 4862-8 + 0-4 4861-5 + 1-6 4959-4 + 0-4 4959-0 + 0-7 5007-2 4-0-4 5007-0 4-0-6 The five lines mentioned above give on an average Sa m = -f 1*1 t.m. A calculated by the third formula is comparable with the wave-lengths of elements, and also with the A derived from the March- April plates (Table VII.). The motion of the new star in the line of sight is here eliminated. If one should succeed in deriving from experiments, or theoretical considerations, the wave-length A' of a certain point a Q of the band belong- ing to a line A' , a! can be calculated from a' = (A' — A / )4500/A / , and the ratio (a Q — a')/4500 is the ratio of the velocity in the line of sight and the velocity of light. The residuals entered in the column headed " Difference " of Table XVII. exceed the quantity that might be expected from the average error of A m , and perhaps the fact that they rest on fewer minima than those of the brighter bands made them less accurate than the latter. Some of the bands call for special remarks. The zero of the second band lies 1*6 t.m. from the calcium line 3968*6, and 3*2 t.m. from the hydrogen line 3970'2. Owing to the good definition of its minima and its isolated position, its wave-length is one of the best determined of the spectrum, and its error is not likely to be greater than the calculated average error. As r>a m cannot be so much in error, I take the band to belong neither to calcium nor to hydrogen. The bright band whose zero has the wave-length 4641*3 cannot be identified. This band overlaps the band, A = 4611 '9, and only two of the three minima were measured. The zero would almost agree with the bright hydrogen line 4633*8 if the observed minima were not the first and second of the standard band, but the second and third ; an assumption which changes the wave-length of the zero by the distance of two minima. This identification is bracketed as an alternative, though it is questionable, because the first calculated maximum of the band has not been observed (see Table XII.). The next band, A = 4687*1, is certainly THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 285 not due to the hydrogen radiation of wave-length 4684. Near A = 4725 , 3 is the hydrogen line 4723 - 6, which in the March-April spectrum was identified with a band Table XVII.— Spect rum of Nova Perseifrom 1901 August 1 to 1903 January. Nova Peksei. Planetary Nebulae. Nova Aurigse. i *0 Q3 O S Degree of Intensity of Maximum of Band on Photo- Plate. Io Relative Intensities of the Radiations. (Campbell). (Campbell). o U 60 a a hO o OS I— 1 1901, October 31 to November 13. 5 o OS o u . O x> |a O £ OS I— 1 o <-§ So o CO CO ^ 1— ( o§ OS s 1-9 O OS O -t-» o> ^ o ,n of OS A I \ I 3868-7 11-0 7-0 0-4 30 24 5 3868 4,5 3967-0 9-5 5-0 0-8 25 18 8 3969 4,5 396 0-5 4102-7 + 0-9 3-1 1-3 0-7 0-8 7 5 4 8 4101-8 5,6 4098 2 4340-2 -0-5 5-2 3-8 30 0-5 11 11 11 6 4340-7 5,6 4336 1 4363-7 10-1 7-7 2-3 20 20 9 4363-8 2,4 4358 8 4611-9 1-4 1-7 0-2 0-3 4 5 2 4 4610 0,1 460 1 4641-3 5-5 3-1 1-0 o-i 11 9 5 2 or (4634-6) 4637 0,2 4630 7 4687-1 5-2 4-0 1-2 0-3 11 11 6 4 4687 2,5 4681 4 4714-8 6-5 2-6 0-5 o-i 13 8 3 2 4715 2,4 471 1 4725-3 5-3 6-0 1-2 11 16 6 4862-0 + 0-5 4-8 1-5 0-3 o-i 10 5 2 2 4861-5 v.b 4857 10 4958-6 -04 2-6 2-0 0-5 0-4 6 6 3 5 4959-0 v.b 4953 30 5006-4 -0-6 4-9 3-5 2-5 10 10 10 10 10 5007-0 v.b 5002 100 Faint Bands. 3813-3835 0-5 6 3889 - 3952 0-2 0-2 1 3 3889 0,4 4027 - 4045 0-8 4 4026 0,4 4063 - 4080 0-6 o-i 0-5 o-i 2 1 3 2 4140-4165 0-5 0-2 2 2 4220-4265 0-5 0-5 0-5 0-3 2 2 3 4 424 4265 0,1 0,1 423 426 1 1 4300-4323 0-8 0-8 0-2 0-1 2 3 2 2 4382-4398 0-8 0-8 0-3 0-3 2 3 2 4 4390 0,4 438 1 4405 - 4488 1-3 1-5 0-7 0-4 3 5 4 5 4472-6 0,5 4466 1 4503 - 4590 0-9 1-0 0-7 0-3 3 4 4 4 4574 4597 4662 0,2 0,1 1,4 451 1 4747-4768 0-4 1-5 0-6 0-1 2 5 3 2 4744 2,4 4776-4786 1-0 0-5 1-0? 4 3 10? 4799-4810 0-6 0-5 3 6 4824 - 4840 0-3 0-6 1 3 4886-4900 o-i 0-5 0-3 3 4 4917-4938 0-1 0-7 0-3 0-3 3 2 4 5031 - 5053 o-i 0-5 0-3 2 2 of medium intensity. As neither the wave-length nor the intensity of the maximum agree with those of the hydrogen line, it probably is not due to hydrogen, though the 286 PROFESSOR L. BECKER ON possibility is not excluded that the March-April identification is wrong, and that both the earlier and the later bands belong to the same radiations. In the second half of the table appear the twelve corresponding lines which Campbell photographed in the spectra of five planetary nebulae, and the range of their intensities in these five spectra, I standing for " feint," and 6 for " very bright." Besides these, there are only two lines, A. = 4662, intensity 1 to 4, and X = 4744, intensity 2 to 4, which Campbell found present in each of the five nebulae. The first falls within the range of the two bright bands X = 4642 and 4688 of the Nova spectrum, and if faint, would be masked by them ; while the second is probably not represented by the faint band 4747 to 4768 of the Nova spectrum lower down in the table. All the prominent lines of the nebulae spectrum are present in that of the Nova, 3868 and 4364, in addition to the principal nebular lines and the hydrogen lines, and their wave-lengths agree within their probable errors. I have already said that the second line of Nova Persei could not be the hydrogen line 3970 '2. The planetary spectrum is not decisive on this point. All the hydrogen lines are bright, and the intensity of 3969 fits into their series, while its wave-length may be a t.m. in error. If it were the hydrogen line, the Nova spectrum after January 1902 could be reconciled with it. It is possible that a faint hydrogen band was superposed on the bright band X = 3967'0, and that after January 1902, when the band faded and the measurements became difficult and less accurate, its principal constituent was the band X = 3970. The only prominent band of the Nova spectrum which has no counterpart in the nebular spectrum belongs to wave-length 4725 '3. 15. Variation of the Bands and of the Radiations in Intensity. — The degree of intensity ^ of the maximum of a band determines the intensity curve of the band. It alone requires to be discussed. From 1901 August 1 to October 6 the observed values of /x agree with each other within their probable errors, and 1 have combined them to mean values. In this period the photo-plates Nos. 8 to 17 were all taken at the same angle of inclination. I have also combined the estimates made in three other periods, using in each period photo-plates taken at angles of 30° and 16°, and discarding those estimates which belong to bands out of focus. The results, which are corrected for the superposed bands, are tabulated in Table XVII. By means of the formulae given in § 9 the relative changes in intensity of the radiations which produced the bands can be calculated. Let a radiation of intensity i in the focal plane of the spectrograph produce in time t a degree n of blackness on a photographic plate whose sensitiveness is s for the wave-length of this radiation, and I designate these conditions by (i, s, m, t), and let a radiation i produce in the same time t on the same plate, for sensitiveness s a degree yw of blackness (%, s , /w , t), and for sensitiveness a' a degree /*'„, (i , s, // , t). I define sensitiveness by st = constant for the same intensity of radiation and the same degree of blackness. i/i is the quantity wanted. I apply formula (5) to (i, s, n, t) and (i , s, /*'„, t). THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 287 * Y = M_ /(/*'o) can be eliminated by /(/*<>)• I a Pply formula (2) to (i , s , n , t) and (i , s, m'o, 0» which can be replaced by (i , s , n' , —), then $0 (U) s /W AA) An) Since the sensitiveness is constant for the same wave-length, I/I suffices for our purpose. It appears in Table XVII. , where it is referred to the intensity of the radiation at X =5007. Compared with the March-April spectrum all the bright bands except the three H- bands, and perhaps the bands X = 4634-6 and 4611*9, have come into existence, or have grown much in intensity. The radiation at X = 43637 is twice as intense as that of the three H-bands, which in the spring were the most prominent bands of the spectrum. The other bands observed in the spring have almost entirely disappeared. None of the radiations faded at a slower rate than the nebular radiation X = 5007. Several radiations began to decrease already in October, and others started in November. The decline of the bands X = 43637 and 4725 in the two months 1901 November 13 to 1902 January 12 is further remarkable. Four radiations faded at the same rate, H s , 4612, 4959, 5007, and perhaps H y , while H 3 certainly decreased at a much greater rate in October. I may mention that I calculated s /s by (14) from estimates of degrees of blackness made on a photograph of the sky spectrum. The values i/i calculated by (15) show that, with the only exception of the radiation at X =3869, that at X = 5007 was the most intense already in August 1901. In October and November 1902 the relative intensities agree better with those of the nebular spectrum than before. It must of course be borne in mind that the spectrum on the last plates was extremely difficult to see, the intensity of the maximum of the band X = 5007 being only of degree 1, and that the figures belonging to that period are only a rough approximation. The trend of the table is certainly to show that the intensities are approaching those of the average nebular state. 16. The Faint Bands. — One may conclude by analogy that the faint bands would be reduced to lines under ordinary conditions. In that case, on account of their breadth, several must be due to multiple lines. On a whole they agree fairly well with the maxima of the intensity curve observed in March and April. Considering the uncertainty of the wave-lengths of these faint bands, about 5 t.m., a convincing proof as to their origin cannot be brought, though it is probable that they are the remnants of the bright spectrum in the first months. It may be mentioned that the hydrogen lines given under B in Table VII. also explain them, provided seven of them be excluded. 288 PROFESSOR L. BECKER ON 17. Last Visual Observation of the Spectrum. — On March 3 1903 I inspected the spectrum of Nova Persei in the focal plane of the spectrograph without using an eye- piece, a method which I usually employed prior to the exposure, to make sure that the proper star had been set on the slit. I saw only one bright spot in the whole range of the spectrum which coincided with the air band at 5004. Several times I gained the impression that there was a faint spot near the place of the magnesium line 4481. The comparison was made in this way, that when the eye had been fixed on the spot the spark was switched on for an instant. 18. Curious relation between A -1 of Four Prominent Lines. — The wave-lengths of the zeros of the brightest bands are 3869, 3967, 4364 and 5007. The oscillation frequencies of the first, third, and fourth zeros almost form an arithmetic series, which, continued to the less refrangible side, gives the wave-length of the helium line D 3 , a line which was measured by others in the spectrum of the new star, and also belongs to the nebular spectrum. In the following table I give the wave-length of the helium line, Keeler's determination of the nebular line, and my determination of the other two lines, reduced to the two nebular lines as standards. The formula A" 1 = 17014-2 + 2957*6» - 5'5?t 2 , n = , 1 , 2 , 3 determines A as entered in the last column. The agreement is perfect. Should this be merely a casual coincidence ? A - Vacuum. A Difference. Calculated A. 3869-2 4364-3 5007-05 5875-87 25837-8 22906-8 19966-3 17014-2 2931-0 2940-5 2952-1 3869-2 4364-3 5007-05 5875-87 19. Similarity of the Structure of the Bands in March-April 1901 and after August 1901. — I add Sa m = + 1.1 t.m. to a m of Table XIV., which reduces them to the same zero as was employed for the March- April bands. Both bands are given in Table XVIII. , and also on Plate II. I include the second minimum of the March-April band, though it rests on only two single measurements in two bands, because it seems to fill up a gap in the order of the minima. The extent of the maximum and the position of the minima agree with each other. There is only this difference, that while the March- April band declines to nothing from a = - 12 to - 73 t.m., and from + 13 to + 56 t.m., the later bands fade abruptly on 6 t.m. Between April and August the ends of the bands have therefore decreased at a greater rate than the central maximum portion. I repeat again that from August 1901 to January 1902 no change took place in the structure, and that the extent of the maximum remained unaltered during 1902. It appears that the spectrum converges towards a nebular spectrum, in which each line is broadened 27 t.m. THE SPECTRUM OF NOVA PERSEI AND THE STRUCTURE OF ITS BANDS. 289 Table XVIII. — The Structure of the Bands in March-April 1901, and after August 1901, referred to \ as zero, and reduced to X = 4500. Structure of the Bands. Reversals. March-April. After August. March-April. After August. .£ m ° ^ . a Intensity. «o Intensity. «o Q * <& «o o "3 "3 T.M. T.M. T.M. *-" G % Jo? "a ° « P MM fc3 £ Number of Plates. T.M. 8 § v 5 c Index Number of Plates. CO — pq.S -73 -55 5 -43 5 6 -31-5 7 -239 " 0* - 22 - 8 * 8 3 -21-8 8 -175 9 -17-8 - 16-6 ) £k 3 } -143 -15-5± 2 1 -14-4 11 J 3 -12 13 - 12-2 -10-4 - 8-3 9 11 9 1 2 - 5-8 - 67 7 - 5-4 - 4-8 7 5 - 53 89 8 to 24 - 3-8 13 - 3-9 10 + 1-2 - 0-3 7 5 + 2-2 5 4 + 1-0 48 9 to 27 + 3-2 12-5 + 2-3 + 49 10 9 + 6-8 + 6-5 7 3 + 8-1 6 4 + 7-8 38 9 to 26 ■f 9-5 12-5 + 9-1 8 + 12-4 + 11-8 7 5 + 12-7 3 3 + 12-4 2 lOandll + 13-0 + 13-0 7 11-5 + 15T 3 + 16-8 10 + 16-9 1 + 20-1 8 + 21-3 + 25-1 7 5-5 + 31-7 3-5 i + 34 2 5 + 36-4 1 + 40-5 J 05 + 56 Sharp Fraunhofer line, 1901, March 18-20 and March 25. "290 THE SPECTRUM OF NOVA PERSEL AND THE STRUCTURE OF ITS BANDS. 20. Results. — 1. The spectrum consists of a line spectrum in which each line is broadened into a band, the broadening being proportional to the wave-length of the line and independent of the element. Tables V. and XIV. give the common structure of the bands. The position of the maxima and of the minima or reversals remains unchanged during the whole period 1901 March to 1902 November. (See Table XVIII. and Plate II.) 2. The intensity curve of the spectrum in March and April is satisfied by the hydrogen and helium lines, some of which vary in intensity during this period. (See Table VII. B, and Plates I. and II.) It is probable that the spectrum is due to hydrogen and helium. 3. From August 1 1901 to the end of 1902 the bands belong to the lines of the spectrum of planetary nebulae, and their relative intensities converge towards those of the average nebular spectrum. (See Table XVII. and Plate II.) Probably the March- April spectrum is also faintly present during the whole period. I wish to acknowledge the help I have received from my Assistant, Mr James Gonnell, who attended to the guiding of the telescope and plotted the curves given in the plates accompanying this paper. Trans. Roy. Soc. Edin BECKER: ON THE SPEC^u Intensity curves of the spectra of Nov calculated- servi 4410- 10YA PE RS EI. Plate I. Vol. XLI. rfarch 3, March 18-20, and March 25 5016 — 1 i ! i i | i i i i | i — i — i — i — f — i — i — i — i — | — i — i — i — i — I i i i t i i — i — i — i — 1 — i — i — i i i r 4600 4700 4800 4900 5000 A RITCHI1 ■■ - "■■ El '.' . I Roy. Soc. Edin. Vol. XLI. BECKER: ON THE SPECTRUM OF NOVA P E R S E I. — plate ii. Ptattl. 1901 March 3. CD P co o 3 3" b. o o ST e -i (0 Intensities 0—N(*J^Ulm>Jooioo"'^ u- .f^OO^tJ^O^^JOOtOi o _ o _ z p * p 3 to 1 M> — o o ~ ft* z o o o _ — - *k - o - — — _ *. I o 3 o _ o — — o — ° — 1 & - o ~ - - ° z - z o - "Z 00 o o ~ £>. Z <x> o o - _z S_z 3 — o - -_ ISJ — o Z o — Z o _ tr» Z ° z o o — z o o ~ ~. ! o CO ■o a> o 3 o o •< P a> a. t= o CD Q. CO 3 CO l-t- «< O c -s <. CO CO o o CO a o s. * . Roy. Soc. Edin, Vol. XLI. BECKER: ON THE SPECTRUM OF NOVA P E R S E I. — plate hi. ngtks A m of zero of bands 3869-5 Curves of observed intensity of 12 bands in the spectrum of Nova Persei. 1901 August to November . abscissae a™ (see a 11) M 3968 4103-4 43410 4364-5 4612-6 46420 4687-i 4726-1 4862-8 4959-4 5007-2 TenDi-metr es I ' ' ' ' I ' ' ' ' I I | I II I II | ' I I I i I I | | I I i I i i i i i i i | i i i i i i -20 -15 -10 -5 ' ' O +5 +10 +15 + l 2Q ( 291 ) XI. — The Histology of the Blood of the Larva of Lepidosiren paradoxa. Part I. Structure of the Resting and Dividing Corpuscles. By Thomas H. Bryce, M.A., M.D. (With Five Plates.) (Read January 18, 1904 ; MS. received March 19, 1904. Issued separately November 19, 1904.) The material for the observations recorded in this paper has been kindly lent to me by Mr Graham Kerr, Professor of Zoology in the University of Glasgow. It consisted of some of his beautiful series of cut embryos, and of some freshly-sectioned material which I stained specially for the purposes of the research. The blood corpuscles of the embryo Lepidosiren are exceptionally favourable objects for the study not only of the morphology of the blood, but also of cell structure. The karyokinesis in the red corpuscles presents features of considerable interest — and the phenomena are presented to the observer on such a scale as to render them almost diagrammatic. In the present paper I shall deal with the structure of the corpuscles and the mitotic phases in the erythrocytes, reserving for a future communication the results of studies on the origin and histogenesis of the elements. Methods. For the study of the dividing red corpuscles I selected a stage in which the embryo was small enough to have permitted perfect penetration of the fixative fluids, and yet sufficiently advanced to have its cells free of yolk. The stage selected was that represented in pi. x. fig. 32 of Mr Graham Kerr's memoir* on "The External Features in the Development of Lepidosiren paradoxa (Fitz.)," a larva twenty-four days after hatching. The embryos chosen had been fixed in sublimo-acetic fluid, and the fixation leaves nothing to be desired. The sections were cut at 10 m, which was rather thick for some points, but the nature of the material, owing to the mass of yolk, did not permit of thin sections. The stain employed was in the first instance iron hsematoxylin, with a counter stain of eosin. It was, however, discovered that even at this early stage several varieties of leucocytes were present in the blood, and for the study of these a stain of methylene blue and eosin, and the mixture of Ehrlich known as Triacid were employed. The best results in some respects were obtained with the first named, especially for the centrosome of the erythrocytes, but for the resting red corpuscles and the leucocytes the methylene blue and eosin gave a finer differential colorisation. * Phil. Trans., vol. cxcii. B. 182, 1899. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 1 1). 44 292 DR THOMAS H. BRYCE ON The two dyes were applied successively, and not in a mixture, according to a method described by Dr Gulland, and communicated to me by Dr Goodall. The sections were stained first for about five minutes with a saturated watery solution of eosin, and then after washing, with a saturated watery solution of methylene blue for two or three minutes. They were then washed and differentiated if necessary in 90° alcohol, dehydrated and cleared in pure xylol. The sections stained by this method are as bright after nine months as they were at first. I. Structure of the Erythrocytes. (a) Resting Corpuscles. The red blood corpuscles are oval biconvex discs, varying in size from 42 to 50 m in length, 30 to 36 m in breadth, and 12 to 15 m in thickness. The nucleus occupies the centre of the disc (PI. I. fig. 1, PL IV. fig. 32). It is also oval in shape, measuring 20 to 27 m in length, 12 to 15 m in breadth, and 9 to 12 m in thickness. (1) Cytoplasm. The corpuscle is surrounded by a delicate membrane. The cell body shows a peripheral ring or band, within which there is a coarse meshwork structure. The meshwork is not very regular, but the thickness of the sections intensifies the appearance of irregularity. The meshes are from 3 to 4 m in diameter. The whole reticulum centres on the nucleus, having a general radial direction from nucleus to periphery. At the nodal points there are strongly refractile granules of considerable size. In some corpuscles the fibrillse of the reticulum in the central nuclear portion of the corpuscle are arranged as parallel running threads between the nucleus and the periphery, but it is not quite clear how far this is a normal appearance. The staining reactions of the meshwork are as follows : — With iron hsematoxylin it is grey, while the microsomes are black (PI. I. figs. 1 and 2) ; with methylene blue and eosin, the meshwork stains bright red and the microsomes are dark red spots ; with triacid it is yellow, and the microsomes stand out as darker yellowish-brown points. In some favourable stainings with the last-named mixture the alveoli had a faint pink tinge. In all the larger corpuscles there is a large vacuole, with structureless contents, showing no differential reaction to any of the stains used. Pound the equator of the cell there is a remarkable band about 3 m in diameter. It forms a complete peripheral ring, when the corpuscle is seen on the flat (PI. I. fig. 1, PI. IV. fig. 32). In the fixed cell its appearance is distinctly fibrillar. The fibrillse run THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA. 293 concentrically ; and though they seem for the most part parallel, there is a considerable amount of apparent crossing and recrossing. In profile views (PL I. fig. 2 and PI. IV. fig. 33) there is to he observed at each end of the corpuscle an area free of reticular formation, but occupied by a number of fine points arranged generally in a straight line. These I take to be the cross sections of what appear to be fibrillae seen on the flat. Such a peripheral ring has been described in the red blood corpuscles of the chick embryo by Dehler,* and in amphibian corpuscles by Nicolas + and Meves.| The latter has demonstrated that in Salamandra the ring is fibrillar, consisting of very fine threads running parallel to one another, or a single unbroken thread developed into a skein in the wall of the corpuscle. This is displaced inwards at the beginning of mitosis, under- goes a loosening, and then disappears as such, its substance being apparently employed for the formation of the achromatic figure. The structure thus described by Meves is evidently of exactly the same nature as the band in the Lepidosiren corpuscles, but he finds no network such as I have described, and the question here arises whether that structure is not a precipitation product. A reticular or meshwork structure has been described in amphibian erythrocytes by a number of authors (Leydig, Frommann, Auerbach, Foa, and others), but it has been variously interpreted. Giglio-Tos § figures a reticulum identically like that I have described, in the erythrocytes of the lamprey. What I have named the microsomes he calls hsemoglobigenic granules. Recently Ruzicka || has represented the corpuscles of Rana as having a reticular structure closely resembling that seen in the Lepidosiren cells. Butschli,1T on the other hand, attributes to the outer portion of the corpuscles in Rana an alveolar structure bounded by a distinct membrane. Within this outer zone is an inner girdle-like zone of finely meshed internal protoplasm, while the central nuclear portion is occupied by a space containing stuctureless enchylema, in which there are radiating tracts of protoplasm. I do not propose to discuss the history of opinion on the structure of the red discs, but I may mention that Rollett ** in a recent paper concludes for an alveolar stroma, while WEiDENREiCH'stt recent observations support Schafer's conclusions (published in Quain's Anatomy), that the contents are fluid and structureless, enclosed by a membrane. In this case I feel no doubt of the existence of a membrane, but reserve is necessary as to the reticulum. It must be noted, however, that I am dealing with young corpuscles. JJ * Archivf. mikr. Anat., Bd. 46, 1895. t Bibliographie anatomique, 1896. I Anat. Anzeiger, Bd. 23, 1903. § Giglio-Tos, Mem. Accad. delle Sc. Torino, T. xlvi., 1896. || Anat. Anzeiger, July 1903, Bd. 23. 1 Protoplasm, etc., English trans., 1894, p. 125. ** Pfluger's Archivf. Physiologie, Bd. 82, 1900. ft Arch.f. mikr. Anat., Bd. 61, 1902, p. 459. H Meves, in a paper published since this paper was written (Anat. Anzeiger, vol. xxiv. No. 18), holds that there is no membrane in the amphibian corpuscles. The peripheral ring of fibrillse is the only structural arrangement in Salamandra, but he states that in Rana there is, in addition, a ' Fadenwerk,' which is collected further round the nucleus, especially at its poles, and he quotes Hensen (Zeitschr.f. uriss. Zool, Bd. U, 1862) as having described in the corpuscles of the Frog a granular material round the nucleus, from which threads pass to the periphery. 294 DR. THOMAS H. BRYCE ON The Lepidosiren corpuscles thus resemble those of Salamandra in the possession of a very distinct equatorial band, but in their reticular structure they seem to correspond more to the description given of the corpuscles of the Frog. Taking all the possibilities into account, I adopt the view that the reticulum is not an artifact, but that it represents a protoplasmic framework. This is possibly alveolar in arrangement, but it is clear that the meshes of the reticulum exceed considerably the limit laid down by Butschli for the true protoplasmic alveoli, and greatly exceed those of the optical reticulum seen in the protoplasm of the leucocytes. The erythrocyte is a much differentiated cell, and the structure described is evidently a secondary one. The whole protoplasm is fibrillar, but the framework is not necessarily fibrous or fixed. I believe rather that it is colloidal. I derive it from a vacuolated condition, in which the active protoplasm (Hyaloplasm) is greatly reduced, and it may well be that an original alveolar arrangement has been lost by the breaking through of mesh walls. The peripheral band must be either the cause or the consequence of the shape of the corpuscle. It disappears when the disc begins to round up for division. This suggests the possibility that the appearance is due to a massing of the mesh walls. Further, in the angular interval between the upper and under layers of the membrane round the equator, there is a space (fig. 2, PI. I.) occupied by the fibrillae of the ring cut across. When the corpuscle rounds up, this space disappears, and the band is replaced by a reticular formation. These considerations, combined with observations on young corpuscles, incline me to the view that the ring may rather be the consequence than the mechanical cause of the shape of the corpuscle, but the matter will come up for discussion again in the second part of these studies, when I am in the position to deal with the histogenesis of the cells. # The question here arises whether the corpuscles which have assumed the biconvex disc shape are capable of division. Besides the corpuscles with oval nuclei, there are others with round nuclei, and a smaller cell body showing a finer reticular structure. These do not assume the disc shape, though they are oval in form. They are found in active division. In the second part of this memoir I shall discuss the relationship between these two forms. Meantime it has to be determined whether both classes of cells are dividing elements. In the later stages of mitosis there is little to distinguish the one class from the other, for all dividing corpuscles are spherical. Variation in the size of the chromosomes would indicate a derivation from a coarser or finer chromatin network, and the round-nucleated corpuscles have distinctly a finer network of chromatin than those with oval nuclei. Direct observation, however, shows that by far the greater number of nuclei showing prophase stages are oval in shape, and between * Meves, in a recent paper cited in the note to page 293, concludes that the band is the cause of the biconvex shape of the corpuscle. His explanation of the mechanism does not seem to apply very satisfactorily to the Lepidosiren corpuscles, but I must postpone a discussion of the question until all the stages in their histogenesis have been worked out. It seems to me that it is only by a study of the developmental stages that the significance of the band or ring can be determined. THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA. 295 the rounded daughter corpuscles, and the biconvex disc- shaped corpuscles, are to be seen all varieties of intermediate stages. These must necessarily be corpuscles either assuming the disc form or rounding up again for division. These intermediate forms sometimes possess two centrosomes ; sometimes no centrosome can be demonstrated. The corpuscles showing early prophase stages of the nucleus always have two very distinct centrosomes, and they are either quite spherical (fig. 8, PL I.) or are oval and often irregular, showing, according to the plane of the section, one or two lateral projections (PI. I. figs. 3 and 4). Corpuscles such as that drawn in fig. 4, PI. I. are frequently met with, in which two very distinct centrosomes are present, although the oval nucleus shows still a coarse network. I believe I am justified in stating that, while it is possible that the corpuscles with vesicular nuclei may not divide, at any rate the smaller disc-shaped ones represent the resting phase of the dividing cells. In the resting stage, if this be so, no centrosome is present in any form in which it can be distinguished from the microsomes. (2) Nucleus. As mentioned above, the nucleus is an oval body. It has a very coarse chromatin network (PI. 1. figs. 1 to 5), with large karyosomes close packed. In a considerable number of corpuscles the nucleus is to all appearance a solid mass of chromatin. The reactions of the nucleus to the various dyes is interesting. In iron hsema- toxylin material the chromatin holds the stain with great persistency, so that the erythrocyte nuclei are still intensely black after all the other nuclei have completely surrendered it. With methylene blue and eosin, the network has a blackish violet colour, quite different from the lighter violet of the nuclei of the leucocytes, and again from the pure blue of the nuclei of the cells of the mesenchyme. The chromatin network again selects the orange from Ehrlich's mixture, and has a golden colour. The alveoli are occupied by a delicate green staining, but no linin threads can be made out. In a successfully stained specimen the chromatin of the mesenchyme nuclei selects the basic dye, and their green colour contrasts with the golden yellow of the nuclei of the red corpuscles. Notwithstanding this behaviour to the dyes, the rounded masses in the nuclei are not true nucleoli, but karyosomes,* or at any rate they are local accumulations of the same substance as forms the intervening bars, and, as later, is uniformly distributed along the spireme thread. (b) Mitosis. As I have already stated, no centrosome is to be seen in any recognisable form in any of the resting corpuscles, large or small. * Of. Pappenheim, Virchow's Archiv, vol. 145. 296 DR THOMAS H. BRYCE ON The first evidence of the onset of mitosis is the formation of a bulging of the central nuclear portion of the corpuscle on one side. In this projection are seen in the vast majority of cases two centrosomes lying side by side, and close to the surface of the corpuscle, and remote from the nucleus (PI. I. fig. 4). Each centrosome is the focal point of far-reaching radiations, which are clearly directly continuous with the reticulum of the corpuscle. They have every appearance of being simply a radially disposed portion of the general network. The centrosomes are not connected directly by intervening fibres. In fig. 4, PI. I. an appearance seen in that, as well as other corpuscles, is suggested. On the left of the nucleus the meshes of the network appear drawn out towards the site of the centrosomes, and the radial fibrillae can be traced far out forming the walls of the meshes of the network. This appearance is transitory. In the next stage (PI. I. fig. 8, PI. IV. fig. 34) the lateral wings have been drawn in, and the corpuscle has become spherical. The radiations are confined to one pole of the cell, the centrosomes remain- ing near together and close to the surface. (l) Structure of Centrosome. The structure of the centrosome varies according to the character of fixation and the manner of staining. In iron heematoxylin sections the appearances depend on the degree of abstraction of the stain. When much of the stain is left, the body is a very large one, and the black colour is even continued out along the radial fibrillse. When the decoloration is carried far, there is a much smaller dark point in the centre of a halo staining red in preparations counterstained with eosin. The fibrillse spring from the circumference of this halo (PI. II. fig. 13). This is clearly an instance of concentric decoloration, and the black spot is not a true centriole in Boveri's # sense. I have not been able to demonstrate a single such centriole or pair of centrioles at any stage of mitosis, but frequently the centrosome has the appearance of a grey spot, containing a group of centrioles. A slightly lobed appearance of a solid centrosome points to the same structure, even though no separate granules are to be made out. The question arises whether this is a ' fragmentation ' of the centrosome (Boveri) * or the true structure. With the other dyes used the centrosome is not so vividly differentiated as with iron hsematoxylin, but in view of the tendency of that stain to mask a finer structure by remaining lodged between the smaller elements, a truer picture is perhaps obtained by their use. In methylene blue and eosin sections the centrosome is a red area occupied by fine granules of the same size as the microsomes, but darker in colour, having a neutral tint — an appearance very possibly due to their being massed together. In the same way in triacid preparations the centrosome is yellow, with brownish yellow granules. * Zellen-Studien, Heft iv., " Uber die Natur der Centrosomen," 1900. THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA. 297 (2) Origin of Centrosome. The origin of the two centrosomes is very difficult to arrive at. I have seen only a very few corpuscles with a localised radial disposition of the reticulum in which there were not two centrosomes, either together in the same, or one in each of the adjoining sections of the series, and these few cases are difficult of interpretation. There is not a single clearly defined centrosome, but an area (fig. 3, PL I.) on which the radii converge. This is occupied by granules in every way similar to the microsomes. The extreme rarity of this stage, if it be a stage, shows that it must be a short-lived one, and that almost immediately the centre of activity is duplicated. It is clear that the centrosome or centrosomes described have no direct relationship to that of the previous division. The resting cells have no centrosome, and no granules distinguishable from the microsomes which can be recognised as centrioles. Further, the new centrosomes do not appear where the old disappear, and therefore, unless on the purely theoretical assumption that the centrioles are scattered in the protoplasm, and though indistinguishable retain their identity,* to become the new focus or foci, the centrosome must be considered to appear de novo. That this is actually the case is strongly supported by instances such as these figured on PL 1. figs. 5 and 6, in which two centrosomes are seen so far removed from one another that it is hardly possible to believe that they have not appeared quite independently of one another. It is remarkable, however, that in later stages, when the centrosomes are very far apart, presumably successors of a stage such as figured (PL I. fig. 6), they are still single. I have seen no multipolar figures, and in such cases there is a suggestion that the chromosomes are forming themselves into two groups round the two asters. I have seen only a small number of such figures, but even the one or two I have seen seem to prove that the two centrosomes may appear independently ; and the fact that the independent centrosomes do not divide and form multipolar figures further suggests the possibility, in the absence of any stage in which a single definite centrosome can be confidently asserted, that the two adjacent centrosomes are independent from the first — that is, as definite stainable and recognisable foci. (3) History of the Nucleus during Mitosis. It will not be necessary to deal in detail with the history of the chromatin as it presents only the well-known evolutions ; a few points only require to be mentioned. The spireme thread is not beaded ; that is, there is no distinction between a linin basis and chromatin granules imbedded in it. The whole thread stains uniformly. In this respect it differs from the thread seen in the prophases in the nuclei of the * Meves, Verhand. anat. Gesellschaft, 1902. 298 DR THOMAS H. BRYCE ON leucocytes. The longitudinal splitting takes place early. The Vs are unequal, with one short and one long leg. The latter in the metaphase is of such length that when all seen in one section it extends round a third of the circumference of the cell. This makes the metaphase figures so complicated that I cannot be certain of the number of chromosomes. In the late anaphases the chromosomes are merged again into a seemingly solid mass of chromatin, which no amount of extraction will resolve into separate elements. The long tails are gradually drawn into the common mass and an oval solid nucleus is formed. In many resting cells, as mentioned above, the nucleus has the same character, and the appearances point to the coarse reticular stage being reached by a sort of vacuolation. Throughout all the phases the chromatin retains the staining reactions described for the resting nucleus. (4) History of the Achromatic Figure. At the stage at which we left the centrosomes when they lay close together, and the corpuscle has rounded up for division, we noticed that there were no direct connecting threads between them. On their outer sides the radiations are strong and join the general reticulum. As the centrosomes draw apart (Plate I. fig. 9) it becomes clear that there are still no fibres directly joining the centrosomes, and that the radiations are stronger on the side of the nucleus. Both at the equator of the spindle figure and where the radiations of the asters meet, the fibres seem to branch and anastomose. I think the appearances are in favour of an anastomosis rather than of a mere crossing of the fibres ; one never sees a loose end at any stage of the process. When both centrosomes are sharply in focus at the same time, the axis of the spindle system is seen to be occupied only by a faint system of branching and anastomosing fibrillae. There is, strictly speaking, no ' central spindle ' spun out between the centrosomes, but only two systems, mainly of mantle fibres, which join one another round the equator (PL I. figs. 11,12). In a cross section of the metaphase figure there is no core of fibres representing a cross-cut central spindle in the heart of the equatorial crown ; only a few fine fibrillae are to be made out. The appearances point, not to any new formation of radiating fibres, but to a con- version, step by step, of the general network into radiating tracts, until it has all been drawn into the opposing systems, and the achromatic figure comes to be placed sym- metrically in the corpuscle. As the daughter chromosomes move apart, the axis of the spindle system is seen to be occupied by loosely-arranged, irregularly-disposed fibres ; and as the anaphase pro- gresses, the ' subequatorial fibres ' (Meves) come out more and more clearly, while the axial system becomes more loosely arranged (PI. II. figs. 13, 14, 15 ; PI. V. fig. 39), until we have a central space traversed by coarse much-branched fibres, and bounded THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOX A. 299 laterally by very distinct fibrillse. These branch, and the branches join at the equator those of the fibres of the opposite astral system, while the outermost threads abut against the cell membrane, and those from opposite poles are seen to meet at the point where it is becoming infolded (PI. II. fig. 15 ; PI. V. fig. 39). There is no thickening of the membrane at the point of infolding. I have shown that at all stages the axial system of fibres is very feebly developed. The contortion of these fibres in the anaphase cannot be due to any ' pushing ' force exerted along them, but rather I believe to the accumulation at, or determination of fluid to the equator of the corpuscle. In some preparations I have seen an actual vacuole occupying the spindle axis, as if the protoplasmic threads had been wholly withdrawn towards the poles. The subequatorial fibres become more strongly marked at this stage, and it is certainly suggested that the lines of force are now directed on the cell periphery, and the picture gives the idea that the force that is exerted by or along the lines of these threads is rather a tractive than a pushing one. The determination of fluid to the equator seems coincident with the passage of the chromosomes to the spindle poles. It is to be noticed that the distance between the spindle poles is distinctly increased at this stage. Stages intermediate between that represented in PI. II. fig. 15 and that shown in fig. 17 are rare, suggesting that once the infolding is produced, the cell division is quickly completed. The subequatorial threads, still attached to the cell membrane at the bottom of the furrow, come to be stretched in a straight line between the spindle poles (PL V. fig. 40), and at a later stage (PI. II. fig. 16 ; PL V. fig. 41) form, with the loose fibres in the axis of the spindle, an hourglass-shaped system of fibrillse. These are grouped apparently in bundles, which contract into the ' mid-body ' when division is complete. This has not the ring form seen in some cells, but is a large single body, probably formed from the smaller single granules on the bundles of threads of the previous stage (PL II. fig. 17). It becomes drawn out into a longish thread when the daughter corpuscles separate from one another (PL II. fig. 18 ; PL V. fig. 43). The centrosome undergoes little increase in size during mitosis. There are no phenomena comparable to the enlargement of the sphere which occurs in dividing ova. In the late anaphases it is drawn out somewhat tangentially, and in the telophases it begins to dwindle. It lies in the hollow of the reconstructing nucleus and is difficult to detect, but in oblique sections it is seen standing out clear of the nucleus ; and in such sections, although I have given much attention to the point, I have not been able to convince myself that it was in any case duplicated (cf. fig. 17, PL II.). In fig. 18, PL II. an appearance suggestive of a division is drawn, but careful examination proved that the radiations were all focussed on one point, and that the appearance was an accidental one, due probably to defective fixation. Similar deforma- tions of the reticulum are met with in other cells removed from the centrosomal area. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 11). 45 300 DR THOMAS H. BRYCE ON I am obliged, therefore, to conclude that, as a recognisable structure, the centrosome disappears completely when mitosis is over, and that, in the absence of any proof that a contained centriole divides in the telophase to survive to the next generation, the centrioles also disappear as such, whether they be twin centrioles or a group of centrioles that could be supposed to persist. Since the demonstration by Wilson,* following the earlier observations of Morgan, that centrosomes arise de novo in the echinoderm egg during artificial parthenogenesis, a conclusion such as I have come to will seem less improbable than it would have some years ago. (c) Interpretation. As stated above, the conclusion was arrived at that the meshwork seen in the fixed corpuscles represented a protoplasmic framework in the living cells. Certain features of the resting cells, and certain appearances observed during mitosis, suggest that the protoplasm is of a specially viscous or ductile nature. The early history of the achromatic figure and of the centrosome preclude the application to this particular case in sensu stricto of either the fibrillar hypothesis (Van Beneden), or of the doctrine of the organic radii (Heidenhain). Both involve a structure of the resting cell which does not exist in the erythrocytes. In the conceptions of Khumbler,! however, I find room for a free formation of the centrosome ; and the interesting feature of this case is, that the theoretical conditions of his model of cell division are fulfilled more closely perhaps than in any hitherto described. The general reticulum is in the resting cell centred on the nucleus. It is under some degree of elastic tension, but the focus of that tension is not a centrosome, and therefore the conditions are not such as represented in Heidenhain's \ model. On the appearance of the centrosomes, the reticulum begins to show a new disposition. It is now centred on these bodies, and round them is converted into radially directed threads. This radial arrangement of the reticulum is probably brought about by the withdrawal of the mesh walls circumferentially disposed into those radially dis- posed to the centrosomes. Apart altogether from the why and wherefore, the centrosomes and their radiations are a manifestation of a tendency of the protoplasm to retract or concentrate itself at two focal points. The first effect of the retraction is the rounding up of the corpuscle ; the second effect is the separation of the centrosomes. When there are two centrosomes at some distance apart (PI. I. figs. 5 and 6), the reticulum becomes converted into a symmetrical aster round each. When they lie close together, the asters are not symmetrical, for between them the protoplasmic material is limited, and is in large measure retracted on to the opposing centres. The progressive condensation or retraction of the threads on the outer sides being thus in excess of that * Arch.f. Entwickelungsmelc, Bd. xii., 1901. t Ibid., Bd. iii., iv., xvii. X Ver. anat. GeselL, Berlin, 1896. Arch.f. Entvrickelunysmek., Bd. i. THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOX A. 301 between the centres, the result necessarily is to cause the centres to separate from one another. Thus the separation of the centrosomes cannot here be clue to the growth of a central spindle between them, but more probably to the contraction or retraction of the astral rays on their outer sides. When the centrosomes have drawn apart to such a distance that the forces of which each is the expression can dominate half the cell, the whole protoplasm arranges itself symmetrically round them, and a position of equipoise is reached. The cytoplasm is already divided into two exactly equal portions, with a neutral zone between. When the nuclear membrane disappears on the side next the spindle, it is noticed that the spindle system is disproportioned. The greater development on the nuclear side is possibly due to the taking up by the two centres of nuclear substance, rather than to an increase in the growth of the astral rays on that side, as has been suggested. It has been a point much discussed, how far the nuclear substance shares in the for- mation of the spindle system. In this case, while the system is formed apparently wholly in the cytoplasm, it seems almost certain that the achromatic substance of the nucleus is also drawn into and divided in it. The arrangement of the spindle fibres in the anaphases is much like that of the outer polar fibres in Boveri's figures of the dividing eggs of Ascaris* but there is no plate at the equator. That the chromosomes are separated by a pushing force on the central spindle is excluded here by the absence of a developed central mass of fibrils. The subequatorial fibres become exposed on the separation of the daughter chromo- somes, and I believe they are related here to the division of the cell body, not in virtue of a pushing or expansive force, as Meves described,t but of a contractive force. In fig. 15, PI. II. the condition of things is pretty clear. In the axis of the spindle system there is a very loosely arranged mass of fibres, with large spaces between the threads, while peripherally, from under the reconstructing nuclei the subequatorial fibres extend towards the equator, and are there continuous with the cell membrane. The threads from the opposite poles meet exactly at the equator on the surface of the cell. At this stage the spindle poles are separated from one another, whether by a determination of the enchylema, or substance from the contracting daughter nuclei, to the equator or otherwise, and the consequence must be to put the longest subequatorial threads, i.e. those reaching the surface at the equator, on the stretch, and if they be of a colloid nature, they will, by their elastic tension, tend to retract on to their centres. Thus we have produced a disposition of the protoplasmic threads, which is roughly indicated in a rude model which I have constructed (text-fig. 1). It is an indiarubber balloon, with a band applied round the equator, to which threads are attached. The threads are brought out through tubes, the inner orifices of which are carried some distance into the interior. When the balloon is inflated through one of the tubes with the threads loose, the result is such as represented in fig. 1 ; when they are drawn tight * Zellen-Studien, Heft 2, 1888. t Arch. f. Entwickelungsmek., Bd. v. 302 DR THOMAS H. BRYCE ON up, on the other hand, the balloon is divided into two (fig. 2). This simple model is not required, of course, to prove that such a system of threads, if contractile, or under elastic tension, and attached to a cell membrane at the equator, will produce, or at any rate initiate, cell division. Heidenhain's or Rhumbler's models show this quite well, but the device described imitates in this one respect, I think, even better what I believe actually occurs in this special case. There is no apparent sign of growth of the cell membrane at the equator, which is one of Rhumbler's secondary factors. When once the furrow is produced it quickly completes itself, because the external pressure is now related to the two centres, and division takes place in the neutral zone between them. That the subequatorial threads Fig. 1. Fig. 2. become stretched out in the axial line between the centres is seen in the photograph PL V. fig. 40, which closely resembles text, fig. 2 representing the model. That the protoplasm has considerable ductility seems to be indicated by the tardy return to the reticular or alveolar condition, and also by the drawing out of the spindle remnant between the daughter cells into a thread of some length. Turning for a moment to alternative hypotheses as to the structure of the corpuscles, I think the idea that the phenomena are to be attributed to lines of strain in a homo- geneous and continuous colloid substance may be put aside. Although the alveolar theory of Butschli is excluded in the strict sense of the term by the size of the alveoli, the protoplasmic framework behaves much as the hyaloplasmic framework does in Rhumbler's theory and the elastic framework in his model. THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA. 303 It would be beyond the scope of this paper to enter on the possible theories as to the changes underlying the retractive phenomena in the protoplasm, but it may be noted that as the centrosome does not enlarge during mitosis, there can be no actual centri- petal movement of the protoplasm on to that body. The same difficulty presents itself if we supposed, with Rhumbler, that the centrosome acts by the abstraction of water from the hyaloplasmic framework, causing it to thicken and condense, unless it were further supposed that the water entered into new combinations in the centrosome, which it is not very easy to accept. The account I have given above is an attempt to explain merely the phenomena as they are presented in the individual case, and does not involve a general theory of the mechanism of mitosis. It seems at first sight radically different from that given by various observers (Wilson, * Teichmann,+ and others) of the appearances in dividing ova, in which the radiations are conceived as manifestations of an actual centripetal move- ment of the hyaloplasm. It may, however, be that the contradiction is one of appear- ance only. The essential factor is the same in both cases — a centripetal condensation of the hyaloplasm. In very fluid protoplasm like that of the ovum, there may well be an actual centripetal movement ; but in very viscous protoplasm like that of the red corpuscles, which are undoubtedly firm and elastic bodies, the condensation may involve only retraction without a flowing movement. If the framework is fixed peri- pherally the retraction would involve increased tension and the rays would become contractile fibrils. Thus no one explanation will apply to all cases ; for if the centrosome and its radiations are the expression of a condensation of the active protoplasm, due to chemical or physical causes, the mechanical results will vary with the consistency of the medium in which such condensation occurs. II. Structure of the Leucocytes. Though it is now well known that in all classes of vertebrates the blood of the adult contains leucocytes of several different varieties, showing very different reactions to various dyes, little is known about the first appearance of the white elements in the blood of the embryo. The stage of embryonic life at which they appear seems to vary. In Lepidosiren the blood is already at a very early stage provided with several different kinds of leucocytes, but in the present writing I shall describe merely the morphology of the different kinds of free cells I have found in the blood and tissues of the embryo, reserving for a future communication the questions regarding the origin of the different varieties, and the interrelation between them. (1) Small Mononuclear Hyaline Corpuscles. This form occurs sparsely in the blood, but more abundantly in the spaces adjoining the posterior cardinal sinus. * Arch. f. Entwickelungsmek., Bd. xiii. t Ibid., Bd. xvi. 304 DR THOMAS H. BRYCE ON It measures 14 to 16 microms in diameter, and possesses a small halo of very delicate protoplasm, which varies in amount from a zone hardly to be made out except under a high power, to a well-defined envelope to the nucleus (fig. 23, PI. III. ; fig. 44, PI. V.). The protoplasm is nongranular, is hyaline in appearance, and even under a magnifica- tion of 1500 diameters it is not possible to make out more than the vaguest suggestion of reticular formation. In methylene blue and eosin preparations it is very delicately stained by the basic dye, while in those tinted with Ehrlich's ' triacid ' mixture it has a faint grey tinge. I cannot with certainty demonstrate a centrosome. The nucleus is round, with a coarsish chromatin reticulum, loosely arranged. It colours violet with methylene blue and eosin, and no part of the nucleus is oxyphil, the linin taking a cold blue tint, while the karyosomes are deep violet. There is some doubt whether in all instances the nucleus is round, or whether there is a notching at one pole. I have observed some such nuclei, and it is obvious that the notching could only be seen if the section passed through a plane at right angles to it, and through the centre of the body. (2) Large Mononuclear Hyaline Corpuscles. This variety occurs more frequently than the last, and is the commonest form seen in the blood stream. It measures 24 to 26 fj. in diameter. The protoplasm varies in amount, but is always merely a narrow zone surrounding the nucleus. In methylene blue and eosin preparations it has a very delicate blue tint, and high magnification reveals a very delicate meshwork, with microsomes at the nodal points, which stain brightly with the blue dye (PI. III. fig. 24 ; PL V. fig. 45). The nucleus is spherical ; the chromatin network is very loosely arranged, and therefore in a section (fig. 24) one sees only rounded bodies with delicate threads radiating from them. These are not true nucleoli or plasmosomes, but karyosomes. So far as I can discover, plasmosomes do not occur in these embryonic nuclei. The staining reactions are interesting. In iron haematoxylin and eosin preparations the karyosomes are black and the general network red, but the chromatin parts more readily with the black stain than the chromatin of the red corpuscles, so that in sections which are suit- able for a study of the latter the white cells are almost purely red. In methylene blue and eosin preparations the karyosomes are deep violet and the network takes a blue shade, but, as in the small corpuscle, there are no purely oxyphil granules. The deep violet blue stands out in strong contrast to the delicate pure methylene blue staining of the protoplasm. In triacid material (fig. 25, PL III.) the network is green, and the karyosomes almost invariably retain some of the acid dye. The colour is sharply distinguished from the golden yellow of the chromatin of the red corpuscles, but also from that of the general mesenchyme nuclei, which stain pure green in preparations which show a yellow tinge in the leucocytes. In cells which show this staining, the chromosomes during division THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOX A. 305 have the same yellow tint as the karyosomes, showing that the material for the chromosomes is at any rate chiefly drawn from them. Though in the great majority of sections the nucleus appears spherical, it is more than probable that corpuscles showing characters identical in other respects, but with a notch at one pole, as represented in fig. 25, are merely cells cut in a plane at right angles to that in which the rounded nucleated cells are cut. I have observed, however, all degrees of notching, from a slight bay to an angular depression, such as seen in fig. 25, or even to a linear fissure reaching to the centre of the nucleus, the two walls of which are in close contact. In sections such as that drawn in fig. 25, PL III. there is found lying opposite the notch a very imperfectly developed centrosome. I have drawn it as a nearly circular darker portion of the protoplasm, which is the ' attraction sphere,' staining like the protoplasm a neutral tint with triacid. At the centre is a slightly darker circular spot, which I take for the centrosome, but it is impossible to make out either a radial structure in the ' sphere ' or rays extending from it into the surrounding protoplasm. In iron hsematoxylin and eosin preparations the same spot in the cell comes out as a homogeneous area, staining of a darker red tint than the rest of the cytoplasm, but I have never seen a darker spot in its centre. There is no doubt that we have here to do with a protoplasmic area, which corresponds to the area to be described in the next variety of leucocyte, showing an active and operative centrosome, with its attraction sphere and rays. In triacid preparations the protoplasm stains of a somewhat indefinite neutral tint, and no granules are ever to be made out. The cell represented in fig. 25, PL III. is certainly a leucoblast, but there is some reason to believe that certain of the cells like that figured in fig. 24, PL III. bear a relation to the Erythroblasts. These are cells of the same dimensions, but with larger karyosomes and a coarser intervening network, and showing a concentric fibrilla- tion of the basiphil protoplasm. They will be dealt with in Part II. (3) Polymorphonuclear Corpuscles. This variety I have named in keeping with the general terminology of blood histol- ogy, on account of the lobed form of the nucleus. This body may, however, have many forms (as seen in figs. 19, 20, PL II. ; 27-29, PL III. ; and photographs 46, 47, 48, PL V.). Sometimes, as in fig. 27, PL III., the superficial appearance is that of a multinuclear corpuscle, but in reality the nucleus of that cell was single, but much lobulated. This group of corpuscles is characterised by the possession of a well-marked centrosome in active operation. They are frequently seen in active diapedesis. Further, they always show, or almost always show, granules in their protoplasm. I shall first describe the centrosome. In fig. 19, which is the same cell photographed in fig. 49, I have drawn the body without filling in the granular cytoplasm. 306 DR THOMAS H. BRYCE ON The centrosome is a large body, which stains a delicate grey in iron hematoxylin preparations, many degrees lighter in tint than the intensely black centrosome of the dividing erythrocytes. In sections counterstained with eosin the body is red. It does not stain, therefore, like the chromatin. In methylene blue and eosin preparations it is very faint, coming only very indistinctly out as a slightly darker area on the faintly bluish-red protoplasm. In triacid preparations it is very distinct (fig. 31, PL III., and fig. 48, PL V.), and has a neutral tint. I have never seen in my material any finer structure within the centrosome, nor can I make out any granules, single, double, or multiple. I have therefore not adopted the term microcentum (Heidenhain), # but have used the word centrosome in Boveri's sense. Pound the centrosome there is a sphere distinctly radiate (as Van Beneden, Heidenhain, and others have described), bounded by a circle of microsomes. This separates the central from an astral zone, into which the fibres of the central sphere pass. There are no outer circles of microsomes concentrically arranged, as described for some cells (Druner). The radii are at first straight, then becoming vavy, they seem to branch and join the general meshwork (fig. 20, PL II.). t In fig. 21, PL II. is represented an amoeboid leucocyte. The extended pseudo- podium is not straight, but wavy, and its axis is occupied by a core of seeming fibrihse passing from the centrosome. The cytoplasm around this central core shows an exceedingly delicate meshwork structure, but it is difficult to be quite sure of this towards the end of the pseudopodium, which is not well defined, and composed of very delicate substance. From the centrosome to the sides of the nucleus there seems to pass a core of seeming fibrillse, represented in the section by the two lateral strands figured. Another wandering cell is drawn in fig. 26, PL IV. It is passing through a space between a number of other mesenchyme cells. The manner in which the polymorphic nucleus is doubled up is interesting. The meshwork and radiations as well as the centrosome are very obscurely revealed in these granular cells by methylene blue and eosin, but it could be made out that at the extremity of the body the mesh- work was drawn out, and that delicate radiations from the centrosome passed into it. Fig. 22, PL II. represents a corpuscle in which the centrosome has divided. The ' attraction sphere ' has apparently enlarged, and is not now bounded by a circle of microsomes. Round the two centrosomes new radiations are developing within the old sphere, which has now the appearance of an extremely fine feltwork, so fine that it is difficult to convince oneself that there is any structure at all. The protoplasm of these leucocytes is basophil when free from granules ; when only a few granules are present, it stains a delicate warm blue with methylene blue and eosin, and shows a very delicate faint meshwork. The granules colour intensely with eosin, are copper red after treatment with triacid, and blacken after iron hematoxylin. They vary in size in the same cell and in * Archivf. mikr. Anat., Bil. xliii. t Gf. Gulland, Jour, of Phys., vol, xix. THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOS1REN PARADOXA. 307 number in different cells. In many they are scattered very sparingly through the protoplasm (fig. 27, PL III.), in others they are closely packed (fig. 29, PI. III.), while in a smaller number the whole cell body is a uniform mass of closely apposed large granules of nearly uniform size (fig. 30, PL III.). These last-described corpuscles may be classified as (4) Eosinophil Leucocytes. Though they have all the appearances of eosinophils described in adult animals, it is not contended that they necessarily belong to the same category. In leucocytes in which the granules are sparingly distributed, their dimensions vary from the minutest particle to bodies 2 to 4 m in diameter. They are highly refractile, and this gives rise to some difficulty in determining their relations to the meshwork. In the drawings I have represented them as seen surrounded by their halo of refraction. Apart from their actual relationships to the alveoli of the meshwork, it is quite certain they are not the nodal points of a reticulum, but are clearly metaplastic, probably minute drops in the protoplasm, which run together to form larger granules. Exactly similar granules are found in the yolk cells, so that it is probable that these in the leucocytes are derived from that source, in which case they could be neither secretory nor excretory products ; but to this question I shall return in the second part of this memoir. There is always a space clear of granules round the centrosome (figs. 29 and 31, PL III.). The nucleus varies greatly in shape, but in its other characters it agrees with that of the mononuclear leucocytes. The simplest form in which it is found is the horse- shoe shape (PL II. fig. 19 ; PL V. fig. 49). Between this and the complicated lobed condition there are all varieties. Sometimes it is ring-shaped (PL V. fig. 47), while sometimes it is formed of quite a number of lobes (PL II. fig. 20 ; PL III. fig. 27). In no case are the lobes detached from one another to produce a multinucleated cell. They are always joined by attenuated portions of the nucleus. The leucocyte figured in fig. 27, PL III. has all the appearance of a polynuclear cell, but careful scrutiny proved that the several lobes were connected together like the two lobes of the nucleus seen in fig. 29 on the same plate. I have found all the varieties of the leucocytes in mitotic phases, but these are few in number at the stages examined compared with the dividing erythrocytes. The polymorphonuclear may divide with the nucleus in the horseshoe form. The general character of the karyokinetic phases is the same as in the erythrocytes, but the chromo- somes arising from a relatively loose and scantier chromatin reticulum are much finer and smaller, and the achromatic structures are of great delicacy. It is interesting to note that, as has been observed in other cases, the centrosome which is so large in the resting phases is reduced during mitosis to a very fine granule, hardly to be demonstrated. As the object of the present writing is more descriptive than theoretical, I do not TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 11). 46 308 DR THOMAS H. BRYCE ON propose to enter into a detailed discussion of all the theoretical points on which my observations bear. The problems relating to the histogenesis of the erythrocytes, and the origin and interrelations of the leucocytes, will be dealt with in the second part of the memoir. Here I shall only deal briefly with certain more strictly cytological con- siderations, and refer to certain conclusions which the contrasting characters of the erythrocytes and leucocytes seem to warrant. It is to be noted that as the close juxtaposition of the blood elements has exposed them to identical conditions both of fixation and staining, the different pictures pre- sented afford a secure ground for comparative study. In the matter of the protoplasm, the material affords examples of all grades between a purely structureless condition and a coarse reticular formation. I have already sufficiently discussed the reticular structure of the erythrocytes, and expressed my belief that it represents an actual disposition of the protoplasm in the living cell. During mitosis it has been demonstrated that this framework is converted into the achromatic figure. In the leucocytes which show a definite but extremely delicate reticulum, this plays the same part in mitosis as the large network of the erythrocytes. The appearances differ in degree only, not in kind, and therefore there is no sharp line to be drawn between the two, whether one accepts the reticular or the alveolar hypothesis. It would seem to be more or less a matter of the relative proportion of active and passive elements, and of variations in consistency. The differential staining of the chromatin of the erythrocytes is a point of suggestive interest, but I am not competent to deal with the questions of cell chemistry involved. While the behaviour to iron hematoxylin might be due merely to differing physical properties, the reaction to the other dyes indicates a chemical differentiation of the chromatin which must be in some way connected with the functions of the corpuscles. The most suggestive of the contrasts observed is that in the characters of the centrosomes. In the erythrocytes, which are passive bodies, in the resting stages there is no centrososome discernible. The body is related only to the mitotic phenomena, and when the kinetic phase is past it disappears as such. Every fact in its history points to its being merely the central point of a cytoplasmic condensation, whatever may be the physical or chemical changes involved. In the leucocytes the centrosome always stains with the cytoplasm. In its full panoply of sphere and aster it is only seen in the leucocytes which undergo amoeboid movements. This fact supports the view that it is related to these movements, and this is actually demonstrated by leucocytes caught in amoeboid movement (fig. 21, PI. II.). That some cytoplasmic activity exists, centred on the centrosome, is clear, but what the nature of the activity may be is another matter. My observations are too few to warrant my going into this question. One would require to see many more amoeboid leucocytes at all stages than I have done to form any opinion on the general question. I put forward the facts I have observed merely THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA. 309 in support of the supposition, that whatever may be the explanation of the outputting of the pseudopodium in the case of the leucocytes, the withdrawal and forward move- ment must be referred to the centrosome, with its sphere and radiations ; and so far as the appearances go, they are in favour of the idea that the centrosome in this case also is the centre of a contractive force. In the conversion of the leucoblast into the leucocyte, the attraction sphere and aster are gradually unfolded as the cell body increases in size. The nucleus is at first central, bub later assumes an eccentric position, while the sphere moves towards the centre of the cell. I would explain this in the same terms as I explained the rounding up of the erythrocyte, and the separation of its centrosomes. The centrosome being here single, however, it comes to a position of equipoise in the centre of the protoplasm. My reading of the structure of the leucocyte is different from that of Heidenhain,* in so far as the radii seem to me to branch and join the general reticulum, which I believe (with the necessary reservations) probably represents an alveolar disposition of the protoplasm, but they act quite like his organic radii, in respect of the movement of the sphere. With regard to the form of the nucleus in relation to the movement of the sphere, my observations, so far as they go, seem to agree with his in matter of fact, but I have not followed out the point in such detail as to follow him into the domain of theory. EXPLANATION OF PLATES. The drawings were done, in everything but the very minute detail, by aid of the camera lucida (Abb^). The lenses employed were the 3 mm. and the 2 mm., both L4 numerical aperture, apochromatic objectives of Zeiss, combined with the compensating oculars 8 or 12. The magnification indicated was ascertained by the stage micrometer. It is rather greater than the magnification given by the com- binations used, the excess depending on the depression of inclined drawing-table beyond the visual distance. The coloured drawings were tinted with the same stains as used for staining the sections. The watery solutions of aniline dyes colour smooth Bristol board very delicately, and permit of a degree of verisimilitude difficult to attain with ordinary water-colours. The photographs were all taken with the 3 mm. 1*4 numerical aperture achromatic objective and No. 4 projection eyepiece, at a distance which gave a magnification of 800, with the exception of fig. 37, in which the magnification is 1500. Plate I. Fig. 1. Section red blood corpuscle in plane parallel to surface of disc, x 1200 d. Compare photo- graph, PI. IV. fig. 32. Fig. 2. Section of same in place at right angles to surface of disc, x ] 200 d. Compare photograph, PI. IV. fig. 33. Fig. 3. First stage of mitosis. Section passes through corpuscle in a plane vertical to its flat face. It is rounding up for division. Possible phase of single centrosome placed in a projection which has risen from centre of disc, x 1200 d. Fig. 4. Similar stage in larger corpuscle. Two centrosomes. x 1 200 d. Fig. 5. Corpuscle with two independent centrosomes. x 1200 d. Fig. 6. Corpuscle with two centrosomes which have appeared separately as in last, x 800 d. * Loc. cit. 3 1 THE HISTOLOGY OF THE BLOOD OF LARVA OF LEPIDOSIREN PARADOXA. Fig. 7. Case of want of synchronism between centrosomal and nuclear cycles. Polar view of corpuscle with nucleus in stage of spireme, while centrosomes still close together x 800 d. Fig. 8. Corpuscle which has rounded up for division. Nucleus in tight spireme stage. x 1200 d. Fig. 9. Drawing apart of the centrosomes. Chromatin thread divided into V-shaped loops, each of which has a long tail-like process from one of its limbs, x 800 d. Fig. 10. Same, a little later, x 2000 d. Compare photograph, PI. IV. fig. 36. Fig. 11. Stage in rotation of spindle system, x 2000 d. Compare photograph, PI. IV. fig. 37. Fig. 12. Fully developed metaphase. x 2000 d. Plate II. Figs. 13, 14, 15. Three stages in anaphase, illustrating especially the emergence of the ' subequatorial fibres.' x 2000 d. With fig. 15 compare fig. 39, PI. IV. Figs. 16, 17, 18. Telophases illustrating division of cell body ; formation of intercellular spindle rem- nant ; reduction of same to form mid-body, which disappears by being drawn out into a thread of some length ; disappearance of centrosome in situ without previous division ; condensation of chromosomes to form a continuous and homogeneous mass. Fig. 16 x 2000 d. ; figs. 18, 19, 1200 d. Fig. 19. One variety of polymorphonuclear leucocyte, showing large centrosome, radiate sphere, and single circle of microsomes, x 1500 d. Fig. 20. Same, with ring-shaped and lobed nucleus. The cytoplasmic meshwork indicated. There are a few minute granules, x 1500 d. Fig. 21. Amoeboid leucocyte, x 2000 d. Fig. 22. Polymorphonuclear leucocyte, showing division of centrosome within old sphere, x 1800 d. Plate III. Fig. 23. Small mononuclear hyaline leucocyte, methylene blue and eosin. x 2000 d. Fig. 24. Large mononuclear cell, methylene blue and eosin. x 2000 d. Fig. 25. Large mononuclear leucocyte, with notched nucleus and centrosome ; Triacid. x 2000 d. Fig. 26. Polymorphonuclear leucocyte ; amoeboid, x 1500 d. Fig. 27. Another form of same, with much-lobed nucleus and scattered small granules. Centrosome not in section. Methylene blue and eosin. x 1500 d. Fig. 28. Another form of same, showing nature of granulation in protoplasm. Centrosome not in section, x 2000 d. Fig. 29. Same, with closely packed granules, x 2000 d. Fig. 30. A further stage of same, showing all the characters of an eosinophil leucocyte, x 2000 d. Fig. 31. Same, showing reaction to triacid. x 2000 d. Plates IV. and V. [All the photographs were taken at a magnification of 800 d.] Figs. 32 and 33. Resting red blood corpuscles. Figs. 34 to 43. Sequence of mitotic phases. Fig. 37 x 1500 d. Fig. 44. Small mononuclear leucocyte. Fig. 45. Large mononuclear leucocyte. Fig. 46. Polymorphonuclear leucocyte, with centrosome and sphere. Fig. 47. Same, with ring-shaped nucleus. Fig. 48. Same, with fine granules. Fig. 49. Same, with large closely-packed granules, i.e. eosinophil leucocyte. The photographs in reproduction have lost much of the delicacy of detail seen in the gelatino-chloride prints. 'logy of the Blood of the Larva of Lepidosiren Paradoxa Part I P] s Roy. S< " •■-•■ ■ £ _ -. ± -■ [ &«% fit 10 12 Bryce Histology of the Blood of the Larva of Lepidosiren Paradoxa Part I Plate n. Roy So XLI. 16 17 18 15 ' 21 19 20 22 Bryce iogy of the Blood of the Larva of Lepidosiren Paradoxa Part I Plate III. iTrani Soc Edm r Vol.XLI. 23 24- 25 26 27 28 • ".•;•■■• 29 30 31 WemerAWmtei lit! Trans- Rov. Soc, Edin. Vol. XLL, Plate IV BRYCE: HISTOLOGY OF THE BLOOD OF THE LARVA OF LEPIDOSIREN PARADOXA PART I. FIG. 32. '"-'■■ FIG. 33. w. FIG. 34. ^^^ Sfc ' I FIG. 35. f ma, FIG. 38. FIG. 36. FIG. 39. FIG. 37. * V FIG. 40. %&AL Trak, Roy. Soc, Edin. Vol. XLL, Plate V. BRYCE: HISTOLOGY OF THE BLOOD OF THE LARVA OF LEPIDOSIREN PARADOXA PART I, m $ FIG. 41. % <t » ■ m FIG. 44. I J FIG. 47. *i r > »>■■• FIG. 42. I » -' , * FIG. 45. v FIG. 48. ** FIG. 46. * FIG. 49. ( 311 ) XII. — The Action of Chloroform upon the Heart and Arteries. By E. A. Schafer, F.R.S., and H. J. Scharlieb, M.D., C.M.Gr. (From the Physiological Laboratory of the University of Edinburgh.) (Communicated March 21, 1904. MS. received August 17, 1904. Issued separately December 14, 1904.) The original design of this research was to determine whether the extract of suprarenal medulla (or its active principle) has the power of antagonising the effects of an overdose of chloroform upon the heart and arterial system. Incidentally the research became extended so as to cover the action of certain other antagonising agents. It further appeared necessary, as the investigation proceeded, to subject the action of chloroform upon the vascular system to renewed study. For although, as the result of numerous recent researches, physiologists are in agreement regarding the general effect of the drug upon the heart, there yet remain various points requiring elucidation both as regards its effect on the heart and on the arteries. Effect of Chloroform upon the Arteries. Singularly little is precisely known as to the effect of the drug upon the arterial system. The most generally received opinion has been that adopted by Bowditch and Minot * to the effect that chloroform exerts, besides a specific action on the heart, a paralysing influence upon the whole vasomotor system, and that the fall of blood- pressure which accompanies its administration is due as well to the dilatation of vessels as to the effect which it produces upon the cardiac musculature. On the other hand, Arloing,! as the result of observations on the rate of flow through the carotid, made by means of the hsemadromograph, inferred that a constriction of arterioles is produced by the drug. Dastre J came to the same conclusion, and referred to it the pallor of the face which is seen in chloroform administration. But it is obvious that a diminution of rate in the carotid might be caused by dilatation of vessels in the splanchnic area, so that these observations cannot be regarded as conclusive. Gaskell and Shore,§ in their cross- circulation experiments, obtained distinct evidence of stimula- tion of the vasomotor centre ; constriction of arterioles and rise of blood-pressure occurring as the result of the action of the drug upon the medulla oblongata. Roy and Sherrington || inferred that constriction of cerebral vessels is produced by chloro- * Boston Med. and Surg. Jour., 1874. Cf. Leonard Hill, Brit. Med. Jour., April 1897. t Thkse, Paris, 1879. J Les Anesthetiques, 1890. § Brit. Med. Jour., 1893, vol. i. \\Jour. Physiol., vol. xi. p. 97, 1890. TRANS. ROY. SOC. EDIN, VOL. XLI. PART II. (NO. 12). 47 312 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE form, while Hurthle* found evidence of dilatation of these vessels, followed by constriction. Newman t observed constriction of pulmonary capillaries in the frog as the result of chloroform inhalation. Sherrington and Sowton,| working with the isolated mammalian heart perfused with Ringer's solution by Langendorff's method, observed a diminished flow of the perfusion fluid when chloroform was added to it ; this they were inclined to ascribe to a contraction of coronary vessels under its influence. C. J. Martin § has suggested that this diminution of flow through the coronary vessels may be accounted for, without the necessity of assuming constriction of those vessels, by the fact that a diminished action of the cardiac musculature, such as chloro- form produces, may tend by itself to diminish the rate of circulation through its vessels. To this we may add that in a heart which is separated from its surroundings and fed by the perfusion of fluid under pressure into the root of the aorta, in which therefore the mechanical conditions are very different from those which obtain normally, the aortic valves do not necessarily act efficiently, but often permit of some passage of fluid into the cavity of the left ventricle, and through this into the left auricle, and so out by the cut pulmonary veins ; and the extent of this valvular defect with the consequent leakage will vary with the condition of tone of the heart and the force of its contractions. Opinions on this subject being thus divided, it appeared important in the first instance to determine what is precisely the action of chloroform upon the arterial system. The method which we have used for this purpose is the classical one of perfusing the vessels with blood or saline fluid containing the drug in solution. The chloroform used for this purpose and in most of our experiments has been Duncan & Flockhart's, sp.gr. 1*49. The result of our preliminary experiments || showed that a solution containing from 1 gramme to 5 grammes of chloroform to the litre of circu- lating fluid produces a marked constriction of the frog's arterioles, and that this con- striction is apparent whether the medulla oblongata and spinal cord are left intact or destroyed. These observations established the fact that for high percentages of chloro- form (5 grammes per litre is approximately a saturated solution, and 1 gramme per litre is therefore one-fifth saturated) there is a pronounced excitation by the drug of the musculature of the arterioles — whether operating directly or through the vasomotor nerve-endings, our preliminary experiments did not decide — -which may contract under its influence to such an extent as almost to arrest the flow of circulating fluid. Since the publication of these preliminary results, C. J. Martin,§ in confirmation of earlier experiments in conjunction with Embley,H has made observations upon the mammalian kidney by the plethysmography method which appear to indicate that in dilute solution — the actual dosage was not determined, but the perfusing fluid (blood) was first passed through the lungs, into which a mixture of air and chloroform vapour was pumped — chloroform has the effect upon the blood-vessels ascribed * Pfliiyer's Arch., vol. xliv. p. 596, 1889. t Jour. Anat. and Phys. vol. xiv., 1879, p. 495. X Thompson-Yates Laboratories Reports, 1903, vol. v. § Private communication. || Communication to the Physiol. Soc. ; Jour. Phys., vol. xxix., 1903. I Brit. Med. Jour., April 1902. lancet, 1902. ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 313 to it by Bowditch and Minot, viz., that of producing dilatation. In repeating and extending our experiments we have therefore included the effect of the perfusion of mammalian organs with various strengths of chloroform dissolved in Ringer's solution. Perfusion of Frog Vessels ivith Chloroform dissolved in Ringer's Solution. The Ringer's fluid used consisted of NaCl, 6 grammes ; CaCl 2 , 0*1 gramme ; KC1, 0*075 gramme ; NaHC0 3 , 0*1 gramme per litre. The chloroform was either dissolved in this solution in proportion determined by weight, or some of the fluid was saturated by being shaken up with and kept over an excess of chloroform, and was assumed to con- tain 1 part chloroform to 200 Ringer, this being the amount water will take up at the ordinary temperature of the air (15° C). This saturated solution was mixed with vary- ing proportions of normal Ringer. A fine cannula having been tied into the bulbus aortse of the frog (R. esculenta or R. temporaria), the fluid was allowed to pass by gravity, at a pressure varying in different experiments from 50 mm. to 150 mm. of water, through the vascular system, and to drip from the extremities of the toes. In our earlier experiments the mode of determining the rate of flow was to count the number of drops per minute ; but this method, although serving to indicate any differ- ences of vascular calibre which are marked, is not sufficiently accurate for slight varia- tions, since the size of the drops is liable to vary somewhat with differences of surface tension of the fluid, and the amount of dissolved chloroform or of intermixed blood and lymph may affect its surface tension. In all later experiments, therefore, the amount of fluid perfusing in a given time was accurately measured. Only the results thus obtained are included in this communication. The result of these perfusion experiments with Ringer's fluid containing dissolved chloroform may be shortly stated as follows : — With the strongest solutions, i.e., from saturated (1 in 200) down to solutions con- taining 1 in 500, a very marked constriction of the arterioles is the result of perfusing with chloroform-Ringer, so that the flow of the perfusing fluid becomes very slow, and may almost cease. With increase of dilution the amount of constriction, as registered by rate of flow, becomes less ; but although very slight when the dilution is consider- able, we have been able to substantiate constriction with solutions as weak as 1 in 20,000. On the other hand, no solution of any strength when perfused through the frog's vessels has given evidence of dilatation of arterioles, the weaker solutions having simply shown themselves inert. If for the chloroform- Ringer which has been passed for some minutes through the vessels, and has produced the diminutions of flow above indicated, normal Ringer be now substituted, the flow again becomes more rapid, but the original rate is rarely again obtained ; in fact, after the chloroform solution has been in action for some minutes, even if the strength of the solution be such as to be insufficient to cause actual constriction of arterioles, there is a tendency towards a gradual diminution in the rate of flow, which appears to be caused by oedema of 314 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE the tissues ; the effect of the chloroform solution on the endothelium of the vessels being such as to render the capillaries more permeable to saline solution. The following experiment will serve as an example of the results obtained with weak solutions of chloroform in Ringer's solution. The numbers represent the amount flowing through the vessels in equal periods of time. Immediately before chloroform perfusion ......... 28 "5 c.c. During perfusion of fluid containing 1 part chloroform in 20,000 Immediately after perfusion of chloroform-Ringer . . .1st period 2nd „ 3rd ,, 22-5 17-5 21 25 Perfusion of Mammalian Vessels ivith Chloroform dissolved in Ringer s Fluid. The kidney, leg, and heart of the cat, rabbit, or dog were employed, and the method of perfusion was the same as for the frog, except that the head of pressure was higher (80 to 100 mm. Hg.). The Ringer solution had the composition: NaCl, 9 grammes; CaCl 2 , 0"24 gramme; KC1, 0*42 gramme; NaHCO s , 0"1 gramme; distilled water, 1 litre, and was warmed to 38° C. by being passed through a glass spiral contained in a water-bath before being conducted to the organ to be perfused, which was itself also kept in a warm chamber at the same temperature. The perfused fluid was either collected in a graduated measure and the amount flowing in a given time recorded, or it was caused to work an automatic " filter," so arranged that every 7 c.c. of fluid produced a see-saw of the filter, and this was recorded by a magnetic signal. In some experiments Ringer's solution, containing a known percentage of chloroform, was, after the normal record of flow had been obtained, allowed to pass for a certain time through the vessels in place of the ordinary Ringer, and was then again replaced by ordinary Ringer, the rate being recorded before, during, and after the passage. In other experiments a chloroform-Ringer of known strength was injected by a fine hypodermic needle through the indiarubber supply-tube of the perfusion apparatus, so as to mix with the inflowing normal Ringer. The amount of dilution of the chloroform-Ringer so perfused was calculated from the amount of fluid flowing through the kidney during the actual time occupied by the injection. This method has the advantages (l) that the chloroform solution only acts for a short time upon the kidney vessels, and is less liable to cause a permanently deleterious effect ; and (2) that the conditions of flow are maintained the same throughout, for if the injection is performed very gradually, no perceptible increase of pressure is caused by it. (It is scarcely possible to change the flow from one vessel to another, as in the ordinary method of testing perfusion, without causing a temporary effect of some kind upon the pressure of the perfusing fluid.) The results yielded by these methods show that in mammalian as in frog's vessels the effect of chloroform solutions of a certain strength is to cause marked constriction of the arterioles and consequent diminution in the rate of flow of the perfusing liquid. If the flow lasts for a short time only, the rate is soon recovered, but prolonged perfusion ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 315 — even with (so-called) normal Ringer, and much more with Ringer containing chloro- form — is accompanied by oedema and a consequent permanent diminution in rate of flow. Kidney. — The strengths of solution which produce constriction in the vessels of the kidney are from saturated (1 in 200) to 1 in 1000 (perhaps somewhat more dilute). But whereas, in the entire frog, solutions weaker than those which cause constriction do not produce dilatation, and have no apparent effect apart from the gradual oedema which results from prolonged action, in the mammalian kidney the effect of weaker solutions is, as C. J. Martin has stated, to cause dilatation. This result has been obtained with solutions of from 1 to 1500 to 1 in 20,000 (in one instance); weaker solutions gave no result. Coronary Vessels. — In employing the heart we have always taken the precaution of tying the pulmonary veins, to prevent loss of circulating fluid by regurgitation past the aortic and mitral valves. The effect of chloroform upon the coronary arteries is to produce constriction in all strengths from saturated to 1 in 10,000. The stronger solutions of chloroform cause so marked a diminution of the rate through the coronary vessels as to almost arrest the flow of fluid ; and this is not due to arrested cardiac action, for on substituting normal Ringer for chloroform-Ringer the rate of flow returns to normal long before the action of the heart recommences. With weaker solutions the effect is also to produce diminished flow, and at no condition of dilution have we obtained evidence of dilatation of vessels. Limbs. — For this purpose the hind limbs of the rabbit and cat have been used. The results are precisely the same as in the case of the coronary arteries of the mammal and the systemic arteries of the frog. Evidence of constriction has been obtained with all strengths from 1 in 200 (which arrests the flow altogether) to 1 in 10,000 (which causes a slight diminution). More dilute solutions are inactive ; we have obtained no evidence of dilatation in these vessels. The following may serve as examples of the results : — Kidney of Rabbit : Amount flowing before chloroform perfusion ........ ,, „ during „ „ (1 in 1500 Ringer) .... ,, ,, after ,, ,, .... ... The same Kidney, later : Amount flowing before chloroform perfusion ........ during „ „ (1 in 700) . . . . Kidneys of Kitten : Amount flowing before chloroform perfusion ........ „ during „ „ (1 in 20,000) , , ,, after ,, ,, ........ In this and the next experiment, as the increase was progressive and there was no return towards normal, it is possible that the increase of rate may not have been due to the chloroform. But it is clear that the drug has not caused constriction of the kidney vessels. 42 c.c. 49 >> 35 5 » 39 C.C 17 5 „ 57 C.C. 60 !! 62 )J 316 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE The same Kidneys, later : Amount flowing before chloroform perfusion 64 c.c. „ during „ „ (1 in 5000) 67 „ „ during „ (1 in 2000) 70 „ Subsequent perfusion with suprarenal produced strong constriction, showing that the arterioles were still active. The effects of stronger and weaker solutions upon the kidney vessels is also well illustrated in the accompanying tracings, which show (in fig. 1) the effect (marked constriction of vessels with rise of pressure Fig. 1. — Perfusion of rabbit's kidney with Ringer's solution. At the time marked by the signal 4 c.c. of the same solution, containing 0*5 per cent, of chloroform, and at the same temperature, was slowly injected into the supply tube. The mixture of this solution with that passing through the tube at the time gave a fluid containing 0*2 per cent. (1 in 500) actually perfused. Notice the rise of pressure due to constriction of the bloodvessels, followed after the passage of the fluid by a dilatation ; also the great diminution in outflow, followed by a slight increase. a, Register of mercury manometer ; b, movements of " see-saw," registered by air transmission : each up or down move- ment represents 7 c.c. of fluid discharged ; c, time in 10 sees. ; d, signal line and pressure abscissa. Fig. 2. — Tracing similar to that shown in fig. 1, but with injection of 4 c.c. of 1 in 1000 chloroform, and actual perfusion of 1 in 6000. Notice the fall of pressure and the increase in rate of discharge, indicating dilatation of arteries. and great diminution of flow) of perfusing a solution of Ringer containing 1 in 500 chloroform ; and (in fig. 2) the effect (dilatation of vessels with fall of manometric pressure, and increase of rate of flow) of passing a solution containing 1 in 6000 through the kidney. Heart of Cat ; Amount flowing through coronary system before chloroform ..... 44 c.c. „ „ „ „ „ during „ (1 in 10,000) . . . 41-5 ., n n n ii i» illlCr ,, . 't't .. ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 317 Another Heart : Amount flowing through coronary system before chloroform ..... „ during „ (1 in 4000) . ,, ,, „ ,, ,, alter ,, ..... Heart of Rabbit : Amount flowing through coronary system before chloroform ..... „ „ during „ (1 in 1500) . ,, ,, ,, ,, ,, aitei ,, . ... The same Heart, later : Amount flowing through coronary system before chloroform ..... „ ' „ „ „ during „ (1 in 1000) . . »» j» jj ? j jj alter ,, ..... The tracing shown in fig. 3 gives an illustration of the effect produced upon the heart by a still stronger chloroform solution, the rate of rjerfusion falling during the passage of the chloroform from 21 c.c. per minute to 5 c.c. per minute, and gradually recovering as the chloroform was washed away. It will be noticed that the recovery of the vessels appears before the contractions of the heart reappear. It can also be seen that the latency of the arterial contraction is longer than that of the heart paralysis which the chloroform produces. 45 c.c. 32 J3 42 >? 58 c.c. 40 ?! 48 i> 48 C.C. 34 n 42 1 ) Fig. 3. — Effect of perfusing 20 c.e. of chloroform-Ringer ( = 1 in 500) through the coronary vessels of the rabbit. a, tracing of manometer connected with supply cannula ; b, register of flow from coronary veins : each interval represents 7 c.c. ; c, time in minutes ; d, signal marking period of injection into supply tube. Note the diminution in rate of flow, and subsequent commencing recovery although the heart remains in a condition of arrest. The rate of movement of the paper is too slow for the individual heart-beats to be seen on the manometer tracing. Hind Limbs of Rabbit : Amount flowing through limbs before chloroform ....... „ „ „ „ during ,, (1 in 10,000) . . . . » )> !> >) alter ,, ....... Hind Limb of Rabbit : Amount flowing through limb before chloroform ....... ,, ,, ,, ,, after passage of 10 c.c. of 1 in 2000 chloroform-Ringer ,, ,, ,, „ in subsequent period ...... These observations show that the kidney differs from the other organs investigated in the fact that the more dilute solutions of chloroform produce an increased flow through the kidney vessels, whereas in the other organs (heart, limbs) the effect of the drug is always in the direction of vasoconstriction. The difference is a remarkable one ; but without discussing it at greater length, we may point out that dilatation of the renal vessels is the normal response of the organ to all but a very few excitants, whereas the normal response of most vessels to an excitation is contraction, and it is possible therefore that the explanation is connected with this difference of " habit " of the kidney vessels as compared with the systemic vessels generally. 86 c.c. 77 j? 84 ») 44 c.c. 39 )? 42 J) 318 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE The action of chloroform upon the vessels when perfused through the isolated organs is a direct action upon the muscular tissue, and not, as in the case of suprarenal extract, upon the terminal apparatus of the vasomotor nerves. This is shown by the fact that apocodeine, which in sufficient dose abolishes the effect of adrenalin,* does not abolish the effect of chloroform in producing vasoconstriction. The following experiment may be quoted in illustration of this statement : — Hind Limb of Rabbit. — Perfusion with 1 in 2000 chloroform reduced the rate of flow during each period of time from 44 c.c. to 39 c.c. After recovery, perfusion with 0"0001 gramme hemisine (adrenalin) added to the normal Ringer brought it down to 10 c.c. After recovery, perfusion with Ringer solution, containing O'OOOl gramme hemisine and - 0075 gramme apocodeine, caused only a slight reduction, soon disappearing. After recovery, admixture with the perfusing fluid of 10 c.c. of chloroform- Ringer ( = 1 in 500), containing 002 gramme apocodeine, caused an almost complete arrest of flow during several minutes. Fig. 4 is a graphic record of this experiment. In all cases the drugs were injected into the tube which supplied the normal Ringer, and the solution became mixed with a certain proportion of this, and warmed to the same temperature by passing through the glass spiral before reaching the organ which was perfused. As a further proof that chloroform acts upon the muscular tissue of the arterioles in Fig. 4. — Effect of injecting 10 c.c. of chloroform-Ringer ( = 1 in 500) containing 0'02 gramme apocodeine through the vessels of the hind limb of the rabbit. a, b, c, d as in fig. 3. The tracing is taken on a more slowly moving surface than that in fig. 3. The initial pressure of the perfusion fluid was lower in this experiment than in the experiment shown in fig. 3, and the supply less free : this, as well as more complete constriction of the arterioles, accounts for the fact that the manometer tracing is much affected in the one case and scarcely at all in the other. producing contraction may be adduced the observation that its action can be got after the neuro-muscular end-apparatus has lost its irritability. Thus in the kidney of a rabbit, which had been killed three hours previously and in which the injection of - 0003 gramme hemisine (adrenalin) produced no effect whatever upon the rate of flow, injection of 20 c.c. of 1 in 200 chloroform-Ringer into the supply-tube reduced the rate from 56 c.c. to 28 c.c. All recent observers are agreed that the fall of blood-pressure which is caused by chloroform is essentially due to its effect upon the heart muscle, the action of which is weakened and eventually paralysed by the drug. Martin and Embley are, as we have seen, inclined to ascribe the fall — in a minor degree — partly to the dilatation which may be produced in the peripheral vessels by small doses of the drug. But since Gaskell and Shore have shown that the effect of chloroform is to excite the vasomotor centre in the medulla oblongata, and thus to cause contraction of the * Dixon, Jour. Phys., vol. xxx., p. 97, 1904. Also Brodie and Dixon, ibid., p. 476. ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 319 arterioles — an observation which Martin and Embley themselves confirm — it is extremely improbable that, while this centre is in action, the dilating effect, if really existent, upon the periphery would be at all apparent. On the other hand, with stronger dosage of chloroform, the direct effect upon the arterioles is always one of constriction. It follows, therefore, that at the beginning of a chloroform inhalation there will be a tendency to counteract the fall of blood-pressure due to heart weakening, by Fig. 5. — End of a fatal chloroform inhalation. Dog, 6570 g. Inhalation through trachea tube of air strongly charged with chloroform vapour. A, blood-pressure curve ; B, line 1 centimetre below zero of blood-pressure ; C, costal respiration (the small waves upon this are heart movements) ; D, diaphragmatic respiration. Time in 10 sees. The signal marks the removal of chloroform. Respirations ceased 20 sees, before the heart. The subterminal rise in blood-pressure which sometimes occurs is shown in this tracing. The increase in size of the manometer excursions is due to a gradual slowing of the heart rhythm, and does not represent an increase of force of the contractions. excitation of the vasomotor centre, and later on, while this may still be active,* a similar tendency to counteract the fall by direct excitation of the peripheral arterioles. As a general rule, the action of the drug upon the heart is so marked as to more than counter-balance the arterial constriction, however produced. But in certain cases a * Reflex constriction of bloodvessels can be obtained, ou stimulating an afferent nerve, even if chloroform anaesthesia is very pronounced, showing that even in deep anaesthesia the vasomotor centre is still active, although its activity is no doubt lessened. Cf. Bowditch and Minot, op. cit. Further, chloroform does not diminish the excitability of the peripheral vasomotor nerves (Scheinensson, Oentralbl.f. d. med. Wiss., 1869, p. 105). TRANS. ROY. SOC. EDIN, VOL. XLI. PART II. (NO. 12). 48 320 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE brief rise of pressure at the beginning of an experiment is sufficiently manifest, and in others again there is apparent, even when the respirations are maintained, a rise towards the end of the experiment. Since in these cases asphyxia is eliminated, and the heart is probably not beating more but less strongly, such a subterminal rise can only be ascribed to the excitatory action of the drug upon the vasomotor centre, or directly upon the arterioles. It is exemplified in fig. 5, which shows the tracing from the latter half of an acute chloroform poisoning, terminating by a slowing and arrest of the respirations, followed after a few seconds, suddenly, by complete arrest of the heart. In this tracing it will be observed that long before the failure of the respiration begins to show itself there is a decided tendency to rise on the part of the arterial pressure, although the heart at this time is not beating more but rather less strongly (the increase in size of _rs_^V_rV_y ^WV.Y'.'.flW|l|\VY<WvW,,.-,,r.~-~-. Fig. 6. — Tracing (dog) showing marked secondary rise of blood-pressure during chloroform inhalation, probably due to early failure of respiration. The chloroform was administered between the two marks on the signal line. Notice the cardiac inhibition, which in this case is more gradually developed than usual, and the subsequent escape of the ventricle, which continued to beat feebly for a minute or two, but with hardly any rise of blood-pressure. Artificial respiration by pump commenced 10 minutes after natural respiration had ceased, failed to effect recovery. a, blood-pressure ; b, tracing from needle passed through chest wall into ventricle ; c, thoracic movements ; d, dia- phragmatic respirations ; e, time in 10 seconds ; /, signal. The horizontal line at b is 10 mm. below the abscissa of blood- pressure. the arterial pulsations seen near the end is a result of slowing of the cardiac rhythm). This rise of pressure therefore must be due to arterial constriction caused by the drug. The chloroform was given as concentrated vapour, producing abolition of corneal reflex in one minute and death in about four minutes ; but how far the constriction was due to direct action upon the arterioles, and how far to an action upon the vasomotor centre, the experiment does not determine. The continuation of the rise in the tracing may perhaps be ascribed to a condition produced by the commencing failure of respiration, the vasomotor centre being stimulated by the venous blood; especially as it is accompanied by a certain amount of cardiac inhibition. Such asphyxial rise may be very marked ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 321 when the respirations become shallow early in the administration, as is shown strikingly in the tracing given in fig. 6. While cases showing such a marked subterminal rise are uncommon, it is not unusual to find a subterminal arrest of the fall of blood-pressure, so that the curve remains for a minute or two at the same level, or shows a more gradual fall than immediately before and immediately after. Such arrest of fall, when unaccompanied by failure of respiration, may also be explained by the constricting action of the drug on the arterioles, acting either through the vasomotor centre or directly. This con- Fig. 7. — Dog, weight 7000 g. Effect of inhalation through trachea tube of air nearly saturated with chloroform vapour. The uppermost tracing (A) is that of the blood-pressure ; the second tracing (B) is costal respiration ; the third (C) abdominal respiration ; the fourth, time in 10 seconds ; and the fifth, the signal marking when chloroform was admitted and stopped. In this experiment the heart failed before the respiration, and about 30 seconds later showed spontaneous recovery, which was, however, only temporary. There was no recovery of respiration. striction, although insufficient entirely to compensate for the continual and gradual weakening and slowing of the heart which is going on the whole time, interferes with the continuous and uniform fall of pressure, which would otherwise show itself. At a much later stage the ventricular contractions, although greatly weakened, produce large fluctuations of pressure in the arterial system, which is then com- paratively empty, owing to the accumulation of blood in the great veins and in the dilated heart cavities. 322 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE Effect of Chloroform upon the Heart. Our experiments abundantly illustrate the fact, which is now unreservedly conceded, that of all ordinary anaesthetics chloroform is the one which produces the most deleterious effect upon both heart and respirations. Nothing is more striking than the comparison of tracings from chloroform experiments, such as those shown in figs. 5, 6, and 7, with one in which ether is the anaesthetic agent (fig. 8). We have further Fig. 8. — This tracing was taken from the same animal as rig. 7, and immediately before it. Air saturated or nearly saturated with ether was inhaled through the trachea tube, between the two points marked by the signal. Previously to this tbe dog was very lightly anaesthetised with chloroform, corneal reflex being present. Compare the effect of ether upon the blood-pressure and respiration with chloroform, in these tracings. investigated the action of chloroform upon the heart in situ with the chest opened, after the method used by M'William ; * and also after removal of the organ from the body, with the coronary vessels perfused by Langendorff's method. f In confirmation of previous observers, we find that the effect upon the organ when its nervous connections are severed, or when the activity of the vagus is abolished by atropine (fig. 9), is to produce a gradual weakening of the contractions (without any marked slowing, although this may appear towards the end of a fatal experiment) Jov,r. Phys., vol. xiii. p. 860, 1892. + Pfluger's Arch , vol. lxi. p. 291,'1895. ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 323 affecting both auricles and ventricles ; but the former are affected much more than the latter. This is also the case when the cardio-inhibitory centre is refiexly stimulated, even in presence of atropine (fig. 10). On the other hand, when the nervous connections are intact, the administration of a strong dose of chloroform not only greatly weakens the force of the cardiac contractions, and thereby causes a marked and progressive fall of blood-pressure, but often, after a primary acceleration, produces gradual slowing of the rhythm, and this is frequently followed by abrupt and complete cardiac arrest (figs. 6 and 7). The abrupt cessation of contraction may affect both auricles and ventricles simultaneously, or the auricles may first stop ; the ventricles, either at once or after a : ■ 1 1 ; : 1 1 1 i i ' ! ' ; ! ' ' "; i :-_;:i 1 1 i :j ! ■ r /■ " .'■!_' ; v : ■ li! !JJj;i t USi) iUMW ' W JJ aiU^O^JLU^JjlU^W JWU*UJUjJ>jijUaa.a>>^^JUU,' i J iilililli'iiiiilN'r, '■[.;(,? :. ''■'■!:--i!ili Fig. 9. — Dog, weighing about 10 kilos. The animal had received some 3 hours previously - 00054 gramme (^1-q gr.) atropine sulphate administered hypodermically. The effect of this was to abolish arrest of the heart on stimulation of the cardio- inhibitory centre (see fig. 10), whilst permitting a diminution in force of the beats, especially of the auricle. It will be seen from these tracings that exactly the same effect is produced in an animal under the influence of atropine by chloro- form alone in strong dose as is caused by reflex excitation of the cardio-inhibitory centre, except that the result is attained more gradually. a, auricular tracing ; b, ventricular tracing ; c, blood-pressure (femoral) ; d, respiratory movements of the thorax, which are continued in spite of the fact that artificial respiration is carried on by perflation ; e, time in 10 seconds ; /, signal. short period of arrest, resuming their action with a rhythm of their own (figs. 6, 7, and 11). The effect entirely resembles that produced by vagal excitation, with the exception that vagal excitation does not, as a rule, by itself produce permanent arrest of cardiac action. But at a certain stage of chloroform anaesthesia the arrest produced by artificial excitation of the vagus may be permanent, or so prolonged as to lead to death. The cessation of the heart's action brings the blood- pressure to zero, and by arresting the 324 PROCESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE Eil Fio. 10. — Testing of cardiac centre by stimulation of central end of one vagus — the other being intact — in the dog from which the tracing given in fig. 9 was taken, and immediately previous to that tracing. a, b, c, d, e, /as before. Notice, as with chloroform, the great diminution in force of the auricular beats, but without arrest or slowing, the diminution in force of the ventricular beats being hardly perceptible. The fall in blood-pressure may be in part due to depressor action, but a similar fall was obtained by stimulation of the peripheral cut vagus. Fig. 11. — Chloroform inhalation. Showing cessa- tion of auricle before ventricle, the latter then assuming its independent rhythm, with larger excursions. a, auricle ; b, ventricle; c, blood-pressure (the alignment of this pen is a little in ad- vance of the others); c 1 , zero of blood-pressnre ; d, time in 10 seconds. Artificial respiration by perflation. Fig. 12. — Effect of excitation (with coil 100) of the peripheral vagus (second signal) during moderately deep chloroform anaesthesia, strong chloroform vapour having just previously been administered during 1^ minutes (first signal). The result is seen to be an immediate heart arrest, with the blood-pressure falling to zero ; the respirations cease 30 seconds after the heart has stopped, but are only gradually arrested. In this case the heart (ventricle ?) begins to escape from the arrest after 40 seconds, beating at first very slowly, but after a minute faster : as the heart recovers, the respirations are also renewed. ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 325 circulation causes also a failure of the respiratory centre. This cause of respiratory failure has been clearly demonstrated by Embley.* The complete similarity between the tracings obtained under the influence of chloroform alone on the one hand (when a concentrated vapour is inhaled), and of ..^^(♦lilllWww.. Fig. 13. — Tracing showing the effect of moderate vagus stimulation (coil 100) during deeper chloroform anesthesia. In this case the excitation was not applied until the respiratory movements had nearly ceased. The effect is to produce complete arrest of the heart, which, however, shows a beginning of escape from the arrest after the lapse of about a minute, and 20 seconds later resumes beating slowly and feebly, and with but little effect upon the blood -pressure. Respirations are not resumed spontaneously, but the animal was recovered 5 minutes after the respirations had ceased, by artificial respiration (compression of thorax) continued during about 2 minutes. The first signal mark shows the period of strong chloroform administra- tion : the second that of vagus stimulation. chloroform plus artificial vagus excitation on the other, shows conclusively that in the former case, as in the latter, the actual cause of the arrest of the heart (and of the Fig. 14. — Tracing showing the effect upon the heart (1) of weak and (2) of stronger vagus excitation during extreme chloroform anaesthesia. The first signal mark shows the period of strong chloroform administration : the second that of weak vagus stimulation : the third that of stronger vagus stimulation. The chloroform was given until the respiration had ceased and the blood-pressure had fallen to 20 mm. Hg. Excitation of the vagus by induction shocks of very moderate strength produced only a momentary arrest of the heart, but stronger excitation caused instant and permanent arrest. respiration as a secondary effect) is inhibition excited through the vagus. Thus figs. 5, 6, 7, and 15 show such a cardiac and respiratory arrest produced by strong chloroform alone, and figs. 12, 13, and 14 the same phenomenon produced under varying degrees of chloroform anaesthesia by vagal stimulation. In figs. 12 and 13 it is seen that the ventricle has escaped from the inhibition and has resumed contraction * Op. cit. 326 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE with an independent (slower) rhythm. In all such cases as these recovery may be effected if artificial respiration be started soon (fig. 15), before the respiratory centre has been allowed to remain too long under the influence of the chloroform plus anaemia ; or if the anaesthetisation be not very deep, there may be spontaneous recovery (fig. 12). In other cases the ventricle does not escape spontaneously (fig. 14) ; but it may be caused to contract by rhythmic compression through the chest wall (fig. 15). The arrest of the heart (and, secondarily, of the respiration) is due therefore to an excitation by chloroform of the cardio-inhibitory centre.* Arloing showed that it does not occur with cut vagi, and this is also emphasised by Embley ; moreover, it does not occur if a small dose of atropine has been previously given (see below, p. 328), and it may also fail to be apparent after prolonged ansesthetisation with chloroform in moderate dosage. It fails to occur also in certain individuals, which seem to be less susceptible than others to the cardio-inhibitory effects of the drug. Instances are shown in fig. 22, A, and — ,\1); .V lj . v " '■" • ■■ •■J .*.' ! ■' J>'-' ' >.*.^ ^i«->^ HH- L i wu^.iwA^-HH- • wmtmtmm Si;;;;; Fig. 15. — Tracing showing (secondary) inhibition of heart from strong chloroform inhalation, with simultaneous cessation ot respiration. Recovery, after 30 seconds' arrest, by means of artificial respiration effected by chest compression. a, respiration ; b, arterial pressure ; c, time in 10 seconds ; d, signal line (2 mm. above abscissa of blood-pressure). also in fig. 16; the latter from a dog in which the inhalation of air strongly charged with chloroform vapour was pushed until respirations had ceased, the heart continuing to beat with great regularity five minutes longer, but during the last three minutes at a slower rate (probably the result of independent ventricular action). In connection with this subject, we have investigated the effect of vagal excitation upon the heart in different stages of chloroform anaesthesia, and the effect of small doses of atropine upon the result of vagal excitation. In light anaesthesia, an adequate stimulation of the vagus produces, as in absence of anaesthesia, complete arrest of cardiac movements with a fall of blood-pressure to zero. But even if the excitation be continued, this condition does not last, for although the auricles may * The above tracings make it abundantly evident that the assertion of Lawrie {Lancet, 1890, vol. i. p. 1393), founded on the report of the Hyderabad Commission, " that sudden death from stoppage of the heart is not a risk of chloroform itself," is completely erroneous. ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 327 remain quiescent, the ventricles escape from the inhibition and resume contracting, at first very slowly, afterwards more quickly, although often somewhat irregularly, so that the blood-pressure again rises. On ceasing to excite the vagus, the heart beats more rapidly and more strongly, and there may be a temporary rise of blood-pressure above the normal average just before the excitation. This characteristic effect is shown in fig. 18. In chloroform anaesthesia the effect upon the heart of vagal excitation is more pronounced and permanent. * The complete arrest of cardiac action may last long enough to cause a concomitant arrest of respiration, and when this occurs, even if the Fig. 16. — Shows a tracing of respiration and blood-pressure under inhalation of air strongly charged with chloroform vapour. This tracing illustrates the type of result obtained when cardiac inhibition does not occur to the extent of causing complete arrest of the heart, but merely a slowing (which may be suddenly increased), the heart failing quite gradually. The chloroform inhalation lasted 4J minutes, at the end of which period respiration had ceased, and was not again renewed until the heart had nearly stopped, when a ' staircase ' group of 25 slow gasping respirations showed them- selves — the so-called 'respirations of the death-agony.' The lid reflex, which was present immediately before the inhalation began, disappeared within one minute. ventricles resume action, their rhythm is very slow, and their force insufficient to raise the blood-pressure much, so that, as a rule, respirations are not spontaneously resumed although artificial respiration may effect recovery. Or it may happen, especially with a strong excitation, that the recovery of the ventricles does not occur at all, and even heart massage, which can be effected by compressing the chest, to which the ventricle may at first respond, may be incapable of causing it permanently to resume its action. * Dastre states that this increase of vagal excitability under chloroform was first noted by Vulpian (0. r. Soc. Biol, 1883, p. 243). According to Fean90IS-Franck (ibid., p. 255), it disappears with increase of anaesthesia, but this is not in accordance with our experience so far as concerns direct excitation. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 12.) 49 328 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE If the chloroform be pushed until respirations cease altogether and the blood- pressure is reduced to a few millimetres of mercury, the only sign of life being the slow beat of the heart, adequate vagal excitation will still cause inhibition, which under these circumstances lasts as long as the excitation ; the removal of excitation being immediately followed by a resumption of the slow, weak beats. Such inhibition can be obtained as long as there is any perceptible beat (fig. 17). We have also re-studied the effect and dosage of atropine in preventing inhibition through the vagus. The results obtained are illustrated in the tracings reproduced in fig. 19. (See also fig. 10 for its effect on reflex vagal excitation.) If a dose of sulphate of atropine of '00002 gramme per kilo, be given hypodermi- cally in the dog, the effect upon the vagus is manifest about fifteen to twenty minutes Fig. 17. — Effect of moderate vagal stimulation in the last stage of chloroform anaesthesia, the respira- tion having long ceased, and the heart beating slowly, feebly, and irregularly. The signal marks the period of vagal stimulation. (The alignment of the signal is a little too much to the left.) It will be seen that excitation of the peripheral vagus still causes arrest of cardiac action, which is at this stage probably entirely ventricular. after administration, and lasts about three hours. The result of such a dose is in some cases to abolish for a time all vagal influence upon the heart (fig. 19, I.). But in most cases, although there is not complete abolition, nevertheless the strongest vagal excitation is unable to produce, in any stage of chloroform ansesthesia, complete cardiac arrest (fig. 19, II. to VI.). There may be, even with comparatively weak excitation, a slowing of the heart and a consequent fall of blood-pressure ; but it is no greater with strong than with weak excitation, and is never accompanied by respiratory arrest, unless in using a very strong excitation there is escape of current to the central end of the nerve. This peculiar condition, in which vagal excitation is unable to cause arrest, but only slowing and diminution in force of the heart, persists for nearly three hours, the ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 329 slowing becoming as time proceeds gradually more marked, and the consequent fall of blood-pressure lower. But cardiac arrest does not show itself until the influence of the atropine has completely passed, and the former conditions can be then restored by a similar dose. There is no reason to believe that the human subject is less susceptible to the influence of atropine than the dog, and the opinion which has been expressed, that an enormous dose would be required to abolish the power of the vagus to cause cardiac arrest, appears therefore to be erroneous.* Fig. 18. — Normal effect, iu the dog, of vagus stimulation of moderate strength (coil 100 mm.) with light anaesthesia, showing the tendency of the heart to escape from the inhibition. This was taken immediately before and from the same dog as the (reduced) tracings shown in figs. 12, 13, and 14, but chloroform was administered in the interval, and the tendency to inhibition is seen in these to be much more pronounced. Since abrupt arrest of the heart and of respirations can be absolutely prevented by prior administration of a small dose of atropine, the conclusion forces itself upon'|us that the precaution of such administration is one that should never be omitted. Atropine cannot, as we shall see, prevent death where a dose of chloroform sufficient to produce paralysis of respiration and complete " paralytic dilatation " of * See on this subject remarks by H. C. Crouch and T. G. Brodie in Trans. Soc. Ancesth., vol. vi. pp. 70 and 81, 1904. J. Harley (Brit. Med. Journ., vol. ii., 1868, p. 320) recommended a dose of from ^g grain to fa grain in man, Dastre (Soc. Biol., 1883, p. 242) states that a dose of atropine amounting to - 0015 gramme ( = fa grain) is sufficient for the purpose indicated. Langi^ois and Maurange (Arch, de Phys., 1895, p. 692) recommend the employment^ oxy-sparteine in place of atropine. III. IV. VI. Fig. 19. — Tracings to illustrate the influence on cardiac inhibition by vagus excitation of a small dose of atropine sulphate ('00054 gramme = T £„ gr.) administered to a dog weighing 28 kilog. =61 J lbs. I., tracing taken 15 minutes after the atropine was administered by injection into the pleural cavity ; II., 30 minutes after ; III., 45 minutes ; IV., 1 hour ; V., 1^ hours ; VI., 2\ hours after administration. The blood-pressure and extent of anssthetisation are approximately the same in all. The strength of stimulus was the same in all, and was adequate to produce strong inhibition in the absence of atropine. Note that this effect is abolished in 15 minutes, and does not reappear in the same form and extent during at least 1\ hours, although there is a gradually increasing amount of inhibition shown as the atropine is becoming eliminated. But even 2£ hours after the injection the strongest stimulus (coil at 0) failed to produce any more effect than that shown in VI. a, blood-pressure curve ; b, respirations ; c, time in 10 sees. ; d, signal showing period of vagal excitation. ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 331 the heart has been administered, but this is a condition which no ansesthetist has any right or occasion to produce. On the other hand, it can prevent the sudden cardiac arrest which may show itself even at a comparatively early stage of chloroformisation, and which may be produced by the drug itself acting on the cardio-inhibitory centre, with (or without) the assistance of an effect reflexly produced upon that centre by the operation which is proceeding ; or by some other concomitant, such as the incidental occurrence of dyspnoea, which is well known also to cause inhibition through the vagus.* Whether the vagi be cut or their terminal branches blocked by atropine, or whether they be left intact, the ultimate effect of chloroform upon the heart, if its inhalation be pushed, is to produce complete arrest in diastole, a condition being observed which has been termed " paralytic dilatation." The condition is a peculiar one, for the cardiac muscle is not only paralysed and incapable of contracting spontaneously, but is in a permanently refractory condition, and incapable of responding to stimuli of any sort.t To all forms of direct stimulation the heart gives no response, although the muscular tissue is not dead, and it suffices to remove the chloroform by producing a flow of unpoisoned blood or circulating fluid through the coronary vessels to restore its rhythmic contractility and its power of responding to artificial stimuli.^ This shows that the refractory condition is due to the influence of the drug upon the heart, and it is commonly assumed by writers upon the subject that chloroform enters into combination with the contractile substance of the cardiac muscle and thereby deprives it of irritability. That this assumption is not justified is clear from the fact that no such effect — in doses which are more than sufficient to paralyse the heart — is produced upon either skeletal or upon plain muscular tissue. It is impossible to believe that the chemical constitution of these forms of contractile tissue is so different from that of heart muscle, that the one combines with chloroform and is thereby rendered devoid of irritability, whilst the others show no tendency to combine with or to be materially affected by the drug. It is much more probable that the effect produced is one of excitation of the terminations of the inhibitory nerves, the heart being thereby rendered irresponsive to stimuli. The argument that may be urged against this hypothesis, that if this were so the effect of chloroform in paralysing the heart would be prevented by atropine, is met by the statement that, although atropine blocks the * An instance of the last-named complication is illustrated in fig. 20. In this animal the breathing was laboured, owing to obstruction of the air-tubes by mucus. There was marked dyspnoea, and the heart-beats were very slow and even arrested whenever the dyspnoea became intense. The violent respiratory efforts succeeded from time to time in clearing the air-passages, and this was followed by partial recovery. This pronounced inhibition was due to asphyxia, which, if more marked than in the instance given, may lead to entire arrest of the heart. Such inhibition from asphyxia does not occur with cut vagi. The condition is one which is not unfamiliar to anaesthetists, who are cognisant of its cause and danger. It is not liable to occur if a prior dose of atropine be administered, partly on account of the effect of this on the vagi and also because atropine tends to prevent the secretion of the mucus which causes the obstruction to respiration. This reason for the administration of atropine will apply equally to ether as to chloroform anaesthesia. t Sherrington and Sowton (op. cit.), in the isolated and perfused cat's heart arrested by chloroform, obtained a renewal of the contractions on stimulation of accelerator nerves. But it is doubtful if this could be obtained with ■a strong dose of chloroform. X Sherrington and Sowton. 332 PROFESSOR E. A. 8CHAFER AND DR H. J. SCHARLTEB ON THE inhibitory path, there is no conclusive evidence that it acts upon the inhibitory end- apparatus in the muscular fibres. According to this view the chloroform-heart — provided that the dose is insufficient to kill the contractile tissues generally— is in a condition of active inhibition rather than in one of passive paralysis. In support of this, it may be stated that although, if the chest be opened immediately after death, the heart may be completely irresponsive to all forms of stimuli, after a little while it often happens that it begins to respond and even exhibits feeble spontaneous contraction, Fig. 20. —Cardiac inhibition produced during chloroform inhalation by dyspnoea resulting from accumulation of mucus in air-passages. A, blood-pressure ; A 1 , line 1 centimetre below the zero of blood-pressure ; B, costal respiration : C, diaphragmatic respiration. The dyspnceic condition is shown by the extreme rapidity of the respiratory movements at the left hand of tracing. About the middle of the tracing the obstruction to the passage of air was removed, and with the disappearance of the dyspnoea the heart resumed its normal rate of rhythm. although the chloroform has not been washed away. The phenomenon may be explained if we assume that the inhibitory end-apparatus has died before the contractile substance of the muscular fibres. To sum up this part of the subject, the conclusions which it appears justifiable to draw regarding the causation of death from the effect of chloroform upon the heart are as follows : — ( 1 ) Death may be caused in the earliest stage of administration by the action of the drug upon the cardio-inhibitory centre, the stimulation being reflex. ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 333 For convenience of description, this may be termed " primary " inhibition. It is prob- able that it occurs but rarely in man, and some altogether deny that it can produce a fatal result ; it is, however, impossible to explain the well-authenticated instances of sudden heart failure at the very commencement of administration without assuming that under certain conditions such primary inhibition, which, when it occurs in other cases, is usually quite evanescent, is occasionally persistent and fatal. # (2) At a somewhat later stage of administration death is liable to occur from sudden heart failure due to inhibition produced by the action of the drug on the cardio-inhibitory centre, aided by its action on the neuro-muscular end-apparatus of the heart itself, and also by increasing venosity of blood caused by failing respiration. The liability to this form of inhibition, which may be termed " secondary " (as well as to that mentioned under 1), can be removed by the prior exhibition of a moderate dose of atropine, which, by diminishing or abolishing the effect upon the heart of excitation of the cardio-inhibitory centre by chloroform, deprives it of the power to produce sudden cardiac arrest. This precautionary measure was long ago suggested, t and all recent work on the subject emphasises the importance of its adoption. That the prolonged administration of chloroform itself tends to diminish its excitatory effect upon the cardio-inhibitory centre in the medulla oblongata is probable from the fact that a dosage of chloroform can be given with impunity at the later stages of a long operation which would be highly dangerous if given at earlier stages. The respiration in these " secondary " cases of inhibition may stop simultaneously with, or shortly before, or immediately after the heart. We have frequently succeeded in effecting resuscitation by artificial respiration in animals, in which both heart and respiration had completely stopped at this stage of poisoning after even a minute or two of cessation of heart-beats, and in two cases as long as three and five minutes respectively after complete cessation ; but in other instances we have failed to obtain recovery after three minutes or more of cessation. The two cases just referred to are of exceptional interest. In the one the animal had been under ether for about an hour (the anaesthetic being inhaled through a Y-shaped trachea tube), when chloroform, at first with considerable intermixture of air, was substituted for the ether. The effect was to produce a gradual fall of blood-pressure from 100 mm. Hg. to about 40 mm., after which both it and the respiration, which was much shallower than under ether, remained nearly constant. After five minutes the lateral air-inlet was cut off, and the dog received a much stronger dose of chloroform. The result of this was immediately apparent in a further fall of blood-pressure, and a slowing and irregularity of the respirations, which ceased altogether about two minutes later, although the heart continued to beat regularly and the blood-pressure was maintained and even rose slightly. About one minute twenty seconds after cessation of respiration the heart suddenly * This mode of producing inhibition has been especially emphasised by Arloing (These, Paris, 1879), who describes the effect of chloroform in producing heart failure in terms very similar to those which we have employed. t Pitha (1861, quoted by Dastre) ; J. Harley, Brit. Med. Joum., vol. i., 1868, p. 320 ; Schafer, Brit. Med. , Joum., vol. ii., 1880, p. 620. Fraser (Brit. Med. Joum., vol. ii., 1880, p. 715), Brown-Sequard (G. r. Soc. Biol., 1883, p. 289), and Dastre and Morat (Lyon Med., 1882, and C. r. Soc. Biol., 1883, pp. 242 and 259) have made a similar -recommendation, but have suggested the addition of morphia, and this combination has often been used (first systematically by Aubert, C. r. Soc. Biol., 1883, p. 626). But morphia is in some ways antagonistic to atropine, and tends by itself to exalt the irritability of the cardio-inhibitory centre. Without atropine it would undoubtedly increase the danger of heart-arrest in chloroform administration. 334 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE stopped and the blood-pressure fell to zero. Artificial respiration (by the pump) was now started, and main- tained for five minutes, and two injections of 4 c.c. adrenalin chloride (1 per mille) were meanwhile made into the pleural cavity without the least result being apparent, the heart remaining quiescent and the blood- pressure at zero the whole time. Intermittent compression of the thorax was now substituted for perflation, and the heart responded to this by a pulsation with each compression, although in the intervals the blood- pressure returned to zero. After continuing the compression for a little over a minute, the heart commenced beating spontaneously and the average blood-pressure rose to 50 mm. Hg. It required, however, another three minutes of artificial respiration before the diaphragm began to act and the intermittent compression could be desisted from for a time ; but even then (although no more chloroform had been inhaled) the respirations again gradually failed and ceased after ten minutes. A second spell of intermittent compression, lasting two minutes, now, however, effected complete restoration. When this was established and the lid reflex had become brisk, the blood-pressure being 110 mm. Hg., chloroform was again administered in strong form. The blood-pressure gradually fell. In two minutes the lid reflex had disappeared, and in another minute respirations had ceased, followed in twenty seconds by complete cessation of heart beat. Five minutes was now allowed to elapse, during which the animal was to all appearance dead. Artificial respiration by chest compression was then recommenced, and two more doses of 3 c.c. adrenalin chloride solution were successively injected. Five minutes after the artificial respiration was commenced and immediately after the final dose of adrenalin, the heart began to beat spontaneously, and the blood- pressure, at first very low, gradually rose in about four or five minutes, during which artificial respiration was maintained, to about 100 mm. Hg. Natural respiration was, however, not again resumed, the medulla oblongata having to all appearance been deprived for too long a time in this instance of blood. In the second dog a lethal dose of chloroform vapour was administered twice. The first time both heart and respiration (the latter ten seconds before the heart) had stopped after three and a half minutes' adminis tration. Half a minute later the chloroform was removed, and 4 c.c. of 1 per 1000 adrenalin chloride solution was injected into the pleural cavity. This produced no apparent effect. Three minutes after cessation of heart and respiration, chest compression was begun. Each compression produced a heart response, and the blood-pressure rose from zero to a few millimetres. After four minutes' chest compression another similar dose of adrenalin was injected into the pleural cavity, chest compression being continued. The blood- pressure then began to recover, the heart now beating slightly more rapidly than the chest compression, but natural respiration (diaphragm) was not resumed until another six minutes had elapsed. As in the last case, however, the natural respirations gradually became shallower and slower again, although no more chloroform was given, and fifteen minutes later ceased altogether, the heart and blood-pressure also becoming weak and low ; the administration at this stage of a decoction of pituitary, and later of another dose of adrenalin, with the idea of restoring the heart's action, produced no visible effect. After two minutes' cessation of respiration (the heart still beating feebly), recourse was again had to chest compression and then to artificial respiration by the pump. This was very soon followed by recovery of heart and blood-pressure, and a few minutes later natural respirations were resumed and maintained, and artificial respiration was discontinued ; recovery was, in fact, complete. (3) In late stages of administration the heart is paralysed by the direct effect of the drug, acting either upon its muscular tissue (as is usually assumed), or (as we believe) by exciting the neuro-muscular inhibitory end-apparatus, and through this rendering the muscular tissue non-excitable. This effect can probably only occur with a consider- able dosage of chloroform in the blood, and the respiratory centre is invariably first paralysed, so that the respirations become slow and shallow and cease before the heart ; the time difference between the cessation of heart and respiration being considerably longer than when the cessation occurs early in the administration. This final effect upon the heart is not antagonised by atropine. The heart is found to be entirely inexcitable, and no treatment is of any avail short of removal of the poisoned blood from the coronary vessels and the substitution of blood free from chloroform. It is conceivable that this substitution might be done by heart massage, or even by com- ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 335 pression of the thorax in attempting artificial respiration ; and, as a matter of fact, Schiff, Batelli, and others * have succeeded in restoring the circulation by heart massage combined with artificial respiration (both in dogs and cats, and in one or two rare cases in the human subject), even although a considerable time (fifteen to twenty minutes) had elapsed after complete cessation of the circulation. We ourselves have not succeeded in restoring an animal after the condition here described appeared to be fully established, and we should be disposed to regard the possibility of resuscitation in such cases as remote.t Antagonising Agents. (1) Atropine. — It is apparent from the results obtained by other experimenters, as well as from our own observations, that the chief danger to be guarded against in the administration of chloroform is the inhibitory influence which it produces upon the heart. As we have already pointed out (pp. 328, 329), this influence can be in great measure controlled by the prior administration of a moderate dose of atropine, at least in so far as the primary and secondary instances of inhibition are concerned, and these are the most dangerous because they are apt to occur without the warning which manifests itself in the case of the final heart paralysis, by the prior arrest of respiration. Atropine is therefore to be placed first in the list of antagonising agents ; a dose of y^-q gr. to 3^ gr. for an average man being administered hypodermically half an hour before the administration of chloroform. (2) Adrenalin Chloride. — The employment of this has been suggested in chloroform poisoning by Gottlieb.;}; In the two instances which we have recorded on pp. 333 and 334, which were attended by an entirely unusual measure of success so far as resuscitation after apparent death from chloroform had occurred, we administered successive doses of adrenalin, injected into the pleural cavity, as part of the treatment. These happened to be the first two experiments of the series undertaken by us, and we were led to ascribe much of the success which attended them to the use of this drug, and formed high hopes of the value of its administration in cases of chloroform poisoning. Subsequent experi- ence showed, however, that adrenalin by itself is of little or no avail to restore a heart paralysed by chloroform, and even in conjunction with other remedial measures — of which the most important is without doubt artificial respiration by chest compression — we are not in a position, as the result of a number of trials, to affirm that it is able materially to contribute to the process of resuscitation. (3) Ammonia Vapour. — Einger § first showed that in the frog's heart ammonia acts as a direct antagonist to chloroform, and may even set in activity a heart which has * For references see M. Bourcart, Rev. m,4d. de la Suisse Romande, October 20, 1903. t This is no doubt the condition referred toby Richet (Diet, de physiol., article " Anesthesie," 1895, p. 523) when he avers that when cardiac syncope occurs artificial respiration never succeeds in effecting restoration ; for the statement does not apply to the syncope caused by the secondary inhibition previously referred to. X Arch.f. Path. u. Pharm., Bd. 37, p. 98, 1896. See also Bibdl, Wien. Hin. Wochenschr., 1896. § Practitioner, vol. xxvi. p. 436, 1881. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 12). 50 336 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLTEB ON THE been arrested by chloroform ; and since it also has a stimulating action on the heart and respiratory centre, it is likely that it may prove useful as a restorative in cases in which the pulse and breathing have not altogether ceased. We have investigated the effect in dogs of causing them to inhale a mixture of chloroform vapour and ammonia, made either by dropping chloroform and ammonia upon the cotton-wool of the inhaling bottle, or by mixing chloroform in definite proportions with alcoholic ammonia, using for this purpose a solution of ammonia in absolute alcohol containing 6 "8 per cent, of Fi«. 21.— Instantaneous heart failure caused by inhalation, at the moment marked by the signal, of air charged with vapour from a mixture of 20 c.c. chloroform and 5 c.c. ammoniated alcohol. A, blood pressure ; B, respirations. The latter continued to show themselves at a slow rate for 3 minutes after the heart had stopped. ammonia, prepared for us by Messrs Duncan & Flockhart. A mixture of chloroform and ammonia vapours, even if it contain a comparatively small proportion of ammonia, is too pungent to be administered in the first instance, the irritation it causes to the sensory nerves of the mucous surfaces rendering it practically irrespirable. And if the proportion of ammonia be considerable, this excitation may result in powerful cardiac inhibition, and the heart may instantly and permanently stop (fig. 21). If, however, ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 337 the proportion of ammonia be less and the animal be already completely under the influ- ence of an anaesthetic, the effect of the addition of the vapour of ammonia or ammoniated alcohol to the chloroform inhaled is strikingly beneficial. The blood-pressure falls but slightly even during a prolonged period of inhalation — the strength of the heart is well maintained — and the tendency to failure of respiration, which is so marked a feature }}MVMM!MUMMUMMMBMUMMMiMMm^mmmm!m!i, WWIWN^^ B UllMlillilUUIlUi 1,1 111 ImiiiiiiitiiiiUiii ■ fAiAf.iiiii.ifiW.ii 'iiViV.ii.ii. Fig. 22. — Tracings showing in a comparative manner in the same animal the difference of effect between inhalation (A) of pure chloroform, (B) of a mixture of chloroform with ammoniated alcohol (9 to 1 ), and (C) of a mixture of chloroform and absolute alcohol (9 to 1). Note in A the rapid fall of blood-pressure and the speedy failure of respiration ; in this case the heart continued to beat after the respirations had ceased. After a respiratory arrest of more than a minute, during which the heart showed strong tendency to inhibition, artificial respiration by chest compression was resorted to (the beginning of this is shown) : in rather more than a minute the blood-pressure rose — the natural respirations were then resumed. Note in B and C the very slow and slight fall of blood-pressure, and the complete maintenance of respiration during the whole time of adminis- tration. In all three cases the air of respiration was charged as strongly as possible, at the ordinary temperature of the laboratory, with the vapour to be inhaled. a, respirations ; b, blood-pressure ; c, time in 10 sees. ; d, signal showing period of administration. In all cases there was distinct corneal reflex immediately prior to the administration, and this disappeared within 1 minute. 338 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE of strong chloroform inhalation, is considerably diminished. The comparative effects of inhalation of chloroform alone and of chloroform plus ammoniated alcohol are shown in the tracings A and B reproduced in fig. 22, and the beneficial results of substituting a mixture of chloroform containing alcoholic ammonia for pure chloroform are illustrated in the tracing given in fie;. 23.* (4) Alcohol Vapour. — In order to determine how much of the beneficial effect of the mixture of alcoholic ammonia with chloroform was due to the alcohol used as a Fig. 23. — Beneficial effect upon blood-pressure, heart, and respiration of substituting ammoniated alcohol and chloroform (1 to 9) for the pure chloroform which was being administered to a dog. a, blood-pressure ; b, heart-beats, recorded by a needle passed through chest wall ; c, respiration ; d, time in 10 sees. Notice the increasing strength of the heart-beats and of the respiratory movements. It is also seen that the dropped heart-beats due to vagal inhibition which were occurring during chloroform alone gradually disappear as the result of adding ammoniated alcohol to the chloroform. vehicle for the ammonia, we next proceeded to investigate the results of using for inhalation the vapour given off from mixtures of chloroform and alcohol. We were somewhat surprised to find that the results were nearly as beneficial when alcohol alone was used, as when alcoholic ammonia vapour was employed. The difference between the effect produced upon blood-pressure and respiration by inhalation of pure chloroform in the one case, and by inhalation of a mixture of chloroform containing 1 part in 10 * The addition of ammonia gas to the chloroform to be used for inhalation was advocated by J. Duncan Menzies {Brit. Med. Jour., vol. ii., 1895, p. 871). ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 339 by volume of absolute alcohol, is exemplified in fig. 22 (A and C). It will be observed that the effect of the addition of 1 part absolute alcohol to 9 parts chloroform is to largely prevent the fall of blood-pressure, which is recognised as being one of the most serious dangers attendant on chloroform inhalation, and at the same time to maintain the respirations at a force and frequency very little less than normal. The administration was made in both cases on the same animal by the same method, and using the same quantities of the solutions. It may also be observed how much more readily recovery takes place after removal of the mixed vapour than after removal of the chloroform. On the other hand, disappearance of the lid reflex occurs a little sooner when pure chloroform is used, but the difference is not great. We have tried other proportions of alcohol and chloroform, but have obtained no i!te %i( te *^mm mm ^m mik liKSS Bite Fig. 24. — Administration by inhalation of air strongly charged with pure chloroform during 3 minutes to dog weighing 11 kilog. which had received (1 J hr. and If hr. previously) two doses of '00027 g. (t^ gr. in all) atropine sulphate. The vagus, tested immediately before this tracing was taken, gave no slowing and only a slight fall of pressure, even with coil at 0. Notice (1) the preliminary rise of blood-pressure due to excitation of vasomotor centre, succeeded by (2) a rapid and regular fall, less steep towards the end ; (3) entire absence of slowing of pulse ; (4) increase of rapidity, but decrease of excursion of respiratory movements, which became irregular, and eventually hardly perceptible. a, blood-pressure curve ; b, respirations ; c, time in 10 sees. ; d, signal of chloroformisation and abscissa of blood- pressure. The heart was still beating 4 minutes later, but the blood-pressure was at zero, and the respirations had wholly ceased. The animal was then subjected to artificial respiration by chest compression, and in 1 m. 40 sees, natural respirations were resumed, and the heart and blood- pressure rapidly recovered. The tracing shown in the next figure was taken prior to this one. better results. Indeed, with a 20 per cent, alcohol chloroform the respirations appeared to be more affected than with the 10 per cent, mixture. The beneficial effect can hardly be due to the mere dilution of the chloroform vapour by alcohol vapour ; moreover, dilution with ether has not this effect, but the result is then practically the same as is obtained with undiluted chloroform. It is therefore to be ascribed directly to the beneficial action of the alcohol on the heart and respiratory centre. We are of opinion that a mixture containing one part by volume of absolute alcohol to nine parts of chloroform should be used when chloroform is indicated as the anaesthetic, since these results show that it is far safer in its action than pure chloroform. There seems reason to believe that the greater safety of the A.C.E. mixture over chloroform depends upon the alcohol it contains, and that the ether is unnecessary ; it may further be noted that the alcohol in this mixture is in needlessly 340 PROFESSOR E. A. SCHAFER AND DR H. J. SCHARLIEB ON THE large proportion.* The beneficial effect of adding absolute alcohol to the chloroform used for inhalation is seen in atropinised as well as in normal animals. The respective effects of administering pure chloroform, on the one hand, and chloroform containing 10 parts per cent, of absolute alcohol, on the other, to a dog weighing 24 lbs., which had previously received two successive doses of '00027 gramme of atropine sulphate, are illustrated by the tracings shown in figs. 24 and 25. A mixture which is frequently used by anaesthetists in place of the A.C.E. is one of ether and chloroform without alcohol ; but from the facts here put forward it would seem better rather to omit the ether than the alcohol. This remark is not to be understood as implying that ether by itself is not a safe anaesthetic — far safer than chloroform, however diluted — but merely that it has not the same antagonising influence as alcohol upon the dangerous tendencies of chloroform. Fig. 25. — Tracing showing the effect of the inhalation of air strongly charged with the vapour from a mixture of 9 parts pure chloroform and 1 part absolute alcohol. The inhalation was continued during nearly 5 minutes. The tracing was begun 10 minutes before that shown in the preceding figure, and is from the same animal (under the influence of atropine sulphate). Notice, as compared with fig. 24, (1) the much more gradual fall of blood-pressure, which even after nearly 5 minutes of administration still keeps fairly high, (2) the effect on the respirations, which are far less influenced than by the pure chloroform, being well maintained during the whole time. On desisting from the inhalation, recovery of blood-pressure was rapid, and the lid reflex, which had disappeared early during the inhalation, was brisk 5 minutes after the chloroform and alcohol mixture had been removed. a, blood-pressure ; b, respiration ; c, time in 10 sees. ; d, signal. POST-MORTEM CONDITIONS AFTER DEATH FROM CHLOROFORM. Although these conditions have been often described, it may not be out of place to add our own experience and observations. Heart. — In all the cases which we examined immediately after death, all the cavities — with, sometimes, an exception for the left auricle — were distended with blood, the right auricle and great veins of the thorax enormously so. The left ventricle always contained a considerable quantity of blood, but rather less than the right ventricle. If, however, the examination were made some little time after death, the left ventricle was always found empty and firmly contracted. This change from the full flaccid condition to the empty firm condition took place in one case within twenty minutes, while in others it did not show itself for forty-five minutes. * Cf. on this subject, Quinquard, C. r. Soc. Biol., 1883, p. 425 ; and Dubois, ibid., p. 441. ACTION OF CHLOROFORM UPON THE HEART AND ARTERIES. 341 Lungs. — The pulmonary arteries are greatly distended with blood. Otherwise the lungs usually present a perfectly normal appearance externally. But in cutting them open we found, in six out of twenty cases examined, a considerable amount of frothy mucus in the bronchial tubes. Abdominal Viscera. — These exhibited marked venous congestion, especially well seen in the liver, which may be greatly swollen and project beyond the thoracic cage* It is thus exposed to some risk of rupture if artificial respiration be attempted by the Howard method. This happened in one of our cases, although we were aware of the danger, and always endeavoured to avoid it by compressing the chest well above the liver. All these appearances are very similar to those which result from asphyxia due to deprivation of air, whether caused by drowning or otherwise. But they are produced independently of any asphyxia caused by paralysis of the respiration by the drug, for they show themselves equally when artificial respiration has been maintained by perflation, and the drug has produced death solely by its action on the heart. Nevertheless, the ultimate effect upon the heart of chloroform and of deprivation of air respectively is strikingly similar. In both cases the final result is a condition of " paralytic dilatation," or, as we have preferred to term it, " inhibition paralysis," in which the heart is absolutely refractory to all kinds of stimuli. In the case of chloroform the exciting cause is doubtless the drug itself ; in the case of asphyxia, it is probably the carbon dioxide which has accumulated in the blood and tissues, t * To observe this condition of the liver and abdominal organs, it is necessary to open the abdomen before the thorax. For if the contents of the latter be first laid bare, and any of the great veins injured, the congestion of the abdominal viscera at once subsides, owing to the escape of blood from their vessels. t Cf. on this subject the Report of the Committee on Suspended Animation, Trans. Med. Chir. Soc, 1904, Suppl., p. 63. ( 343 ) XIII. — Continuants resolvable into Linear Factors. By Thomas Muir, LL.D. (MS. received August 22, 1904. Read November 7, 1904. Issued separately January 13, 1905.) (1) It is known that a continuant whose three diagonal are formed of certain equidifferent progressions is resolvable into linear factors, the earliest specimens placed on record being those of Sylvester and Painvin.* The object of the present paper is to show that there are continuants of quite a different type which are also so resolvable, and to expound a general mode of investigating the subject. (2) The continuant of the n tk order ivhose main diagonal is a, a + 2-V-c, « + 2-2 2 -c, a + 2-3 2 -c, and whose minor diagonals are 2-(n-l)b, («-2)(6 + c), (n-3)(6 + 2e), n(b-c), (n + !)(&- 2c), (» + 2)(&-3c), is equal to the product of the n factors {a + 2(n-l)b} ■ {a + 2(?*-3)6 + 2(2w-3)c} • {a + 2(n-B)6 + 4(3n-5)e} • {o + 2(n-7)6 + 6(2«-7)c} ■ {a-2(n-l)& + (2n-2)lc} Taking for the purposes of illustrative proof the case where n = 5, viz. a 246 5(6 -c) a + 2c 3(6 + c) 6(6 -2c) a + 8c 2(6 + 2c) 7(6 -3c) a+18c 1(6 + 3c) 8(6 -4c) a + 32c and performing the operation col x + col 2 + col 3 + col 4 + col 5 we find we can remove the factor a + Sb and write the cofactor in the form a-86 + 2c 3(6 + c) -26 -12c a + 8c 2(6 + 2c) -86 7(6 -3c) a + 18c 6 + 3c -86 . 8(6 -4c) a + 32c Performing now on this cofactor the operation colj + 4 col 2 + 9 col 3 + 16 col 4 * [Sylvester, J. J.] " Theorerne sur les determinants de M. Sylvester," Nouv. Annates de Math., xiii. p. 305. Painvin, . " Sur un certain systeme d'equations lineaires," Journ. de Liouville, 2 e ser., iii. pp. 41-46. Muir, Thomas. " Factorizable Continuants," Trans. S. Afr. Philos. Soc, xiv. pp. 29-33. (I) TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 13). 51 344 DR THOMAS MUIR ON we obtain (a + 46+1 in) 1 36 + 3c 4 a + 8c 26 + 4c 9 lb -21c a+l&c + 3c 16 . 86 -32c a + 32c of which the determinant factor is reducible to the three-line form a- 126 -4c 26 + 4c -206 -48c a+18c 6 + 3c -486 -48c 86 -32c a + 32c The next operation gives in like manner colj + 6 col 2 + 20 col 3 (a + 20c) 1 26+ 4c 6 a+18c 6 + 3c 20 86 -32c a + 32< or (a + 20c) : a -126 -6c 6 + 3c -326 -112c a + 32c and finally the operation colj + 8 col 2 enables us to change this two-line determinant into (a -46+ 18c) 1 6 + 3c 8 a + 32c or (a-46 + 18c)(a-86 + 8c) The desired result thus is (a + 86)(a + 46 + Uc)(a + 20c)(a - 46 + 18c)(a - 86 + 8c) (3) The continuant of the n th order whose main diagonal is a, a + 2(l-3)c, a + 2(2-4)c, a+2(3-5)c, . . . and whose minor diagonals are {n-l)b, («-2)(6 + c), (»-3)(6+2c), . . . . (n+2)(6-3c), (n + 3)(b-ic), (ra + 4)(6-5c), . . . is equal to the product of the n factors {a + 2(«-l)6}, {a + 2(w-3)6 + 2(2n-l)c}, {a + 2('/«-5)6 + 4(2ra-3)c} , {a-2(n-l)6 + 6(«-l)c} . (II) This is established by proceeding in the same way as in § 2, the sets of column - multipliers now being 1, 2, 3, 4, 5 1 , 4, 10, 20 1, 6, 21 1, 8 1 instead of 1, 1, 1, 1, 1 1, 4, 9, 16 1 , 6, 20 1, 8 1 CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. 345 and the resulting factors greater by respectively. 0, 4c, 8c, 12c, 16c, (4) Changing a into a + 2c, n into n— I, b into b + c in § 3 we have a+ 2c (n-2)(b + c) . . . . . . (n+Y)(b-2c) a + 8c (n-8)(6 + 2c) . . . . . (» + 2)(6-3c) a+18c (?i-4)(6 + 3c) . . . . (» + 3)(6-4c) a + 32c . . . . = {a + 2(n - 2)6 + 2(» - l)c} ■ {a + 2(n - i)b + 6(» - 2)c} • {a + 2(»-6)6+10(»-3)c} But the continuant here is the complementary minor of the element in the place ( 1 , 1 ) of t-he continuant in § 2. Consequently by division we obtain zMtlW (n-2)(n + l)(b-2c)(b + c) a+2-l 2 -c ffl + 2-2 2 -c \ h(2n-2)(b-nr. + c)( b + nc- 2c) a + 2-(n-l) 2 -c {a + 2(«-l)6}{a+2(n-3)6 + 2(2n-3)e}{a + 2(n-5)6 + 4(2n-5)e} . . . ■ i\l\\ {a + 2(n-2)6 + 2(»-l)c}{o + 2(n-4)6 + 6(n-2)c} . . . . (5) If in the results of §§ 2, 3 we annex f as a factor' to every term on both sides that is independent of a, the identity is not interfered with. . . . (IV) For, taking (in the fourth order, for shortness' sake) the continuant dealt with in § 2, and putting ct/f for a we have 3b 6(6 - 3c) ± + 6c 2(6 + c) 7(6 -4c) il+16c b + 2c 8(6 -5c) 4 + 30c = (± + 66 V± + 26 + UcY^- - 26 + 20cY^ - 66 + 18c) , whence on multiplying both sides by/ 4 there results a 36/ 6(6 -3c)/ a + 6c/ 2(6 + c)/ 7(6 -4c)/ a + 16c/ (6 + 2c)/ 8(6 -5c)/ a + 30c/ = (a + Qbf)(a + 26/+ 14c/)(a - 26/+ 20cf)(a - 66/+ 18c/) , as asserted. 346 DR THOMAS MUIR ON (6) The sum of the elements of the main diagonal of either of the continuants in §§ 2, 3 is equal to the sum of the factors into which the continuant is resolved. . (V) This is true of any continuant of the form a + x Pi </! a + x 2 p 2 q 2 a + x 3 that is resolvable into factors linear in a. By way of proof we have only to note (1) that since the diagonal term is the only term of the continuant that contains either the n th or (n— l) th powers of a, it follows that the coefficient of a n ~ x in the continuant is x 1 +x 2 + x s + . . . , or 2a? say : and (2) that if a + n-y , a + M 2 , a + M 3 , • • • be the factors into which the continuant is resolved, the coefficient of a" -1 in their product is /«! + m 2 + m 3 + • • • , or 2m say. We thus have 2.r = 2yu,, and . *. na + 2.c = na + 2/x , as was to be proved. (7) The full table of multipliers used in § 2 is found to be 1,1,1,1, l, ,1 1, 4, 9, 16, ,L-C ril 1, 6, 20, ,yC r+li3 1, •1 C r+3, 7 — in other words, each multiplier is of the form r n — ^r+s-1 ) 2s-l , and the question next arises whether the continuant resolved in § 2 is the only one which this set of multipliers is capable of dealing with. In order to make suitable answer we have to ascertain the relations which must exist between the twelve quantities in the continuant &, P'2 > Pi ) Hi P, q, r.s 7i. 72 ' 73 . 74 a 24ft 5 7l a + p 3/J, 6 7'2 a + q 2(3 3 7y 3 a+r ft 8y 4 a + s in order that it may be resolvable into linear factors by means of the operations of § 2. CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. 347 The performance of the first operation on the columns evolves the equations n + 8f3 l = 5y, + a +p + 3/3 2 , = Qy 2 + a + q+2f3 3 , = 7y 3 + a + r + (3 i , = 8y 4 + a + s . Then the factor a + 88 x being removed, the cofactor becomes expressible as a determinant of the next lower order, viz. a + p-8/3, 3/? 2 6 yi -8^ a + q 2/3 3 . -8/3, 7y 3 a + r /3 4 - 8/3, . 8y 4 a + s and as by hypothesis the performance of the orjeration colj + 4 col 2 + 9 col 3 +16 col 4 enables us to remove the factor a + p — 88 t + 12/3 2 we obtain three other conditional equations, to be followed at the next stage by two others, and last of all by one. Of these 4 + 3 + 2 + 1 equations the four first obtained are obviously best used to determine p, q, r, s in terms of the B's and y's, the results being p = 8/3 x - 3/3 2 - 5 Yl , q = 8/3, - 2/3 3 - 6y 2 , r = 8/3, - ft - 7y 3 , a = 8/8, -8y 4 . The remaining six form a very interesting set : after simplification they are 12ft - 1 8/3, + 5/3 3 = -10 yi + 9y 2 64/3, - 81/3 2 + l(3 i = - 45 Tl + 35y 3 15/3,-18^ = -10y, +7y 4 45/3 2 - 60/3 3 + 1 4/3 4 = - 36y 2 + 35y 3 24/8 a -25/8, = -15y 2 +14y 4 P% " Pi = - Y3 + Y4 • Taking the first three and using with them the multipliers 7, — 1, 1 respectively, we find, on adding, that 5(& + Yi) - 9(/3 2 + y 2 ) + 5(/3 3 + y 3 ) - ( 4 + /3y 4 ) = 0; similarly from the subset of two there results by subtraction 3(/3 2 + y 2 ) - 5(/3 3 + y 3 ) + 2(/3 4 + y 4 ) = 0; and the final set, of course, is By means, therefore, of these three derived equations we arrive at the proposition that in the determinant under discussion the sum of any 8 and the corresponding y is constant. This being equivalent to only three equations, and other three being still un- accounted for, we put Yl > 7-2 . Jz . Ji = °" - Pi > & ~ P-i > °" ~ As » °" _ Pi < 348 DR THOMAS MUIR ON and learn (1) that one of the equations is not independent of the others, (2) that the /8's are connected by the equation i e i -2i8 2 + ^ s = 7(&-2/3 3 + /? 4 ), and (3) that a- is expressible in terms of any three of the ,8's, for example, o- = - 2/J, + 9/3 2 - 5/J s . The conclusion thus is that in the continuant with which we started we can retain any three of the /8's , and express in terms of these the fourth /8 , all the y's , and p , q , r , s , — thus obtaining a function of four variables which is resolvable into linear factors. (8) Had the determinant operated upon been of the sixth order, we should still have found 0- = - 2/^ + 9,8 2 - 5/8 3 and the first four /8's connected by the same equation as in the preceding case, but there would have been a fresh equation of condition connecting the second set of four consecutive /8's, viz. /3 2 -2/} 3 + &-= 3(/S 3 -2/3 4 + /3 5 ). Similarly the case of the seventh order would be found to differ from that of the sixth merely in having the additional equation 5(/3 3 -2 i Q 4 + /3 5 )= Il(j8 4 -2ft + j8 a )j and so on. As the result of all this we therefore affirm that — If the continuant a 2(?i - l)/5j ny 1 a + ]> (n - 2)/3 2 (w+l)y 2 a + q (?i-3)/3 3 (?i + 2)y 3 a + r be resolvable into linear factors by means of the set of multipliers llill 1 4 9 16 1 6 20 1 8 1 then (1) every four consecutive /3's are connected by a linear relation, viz. l.(ft-2/3 2 + /3 3 ) = 7-(/J 2 -2/3 8 + &), 3.(/? 2 -2& + &) = 9-(/3 8 -2/3 4 + /5 6 ), 5-( i 8 3 -2/8 4 + /3 5 )= ll.(/3 4 -2/3 5 + /3 6 ), i/iws making all the fi's expressible in terms of any three; (2) all the y's are expressible in terms of the same three /8's because of the fact that fi m + y m = — 2/3 l + 9/8 2 - 5@ 3 for all values ofm; and (3) p, q, r, . . . are also so expressible because the sum of the elements of any row of the continuant is constant . .... (VI) CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. 349 (9) Instead, however, of taking a and three of the /3's as variables it is better to take a and ^ , /3 2 — /3 2 , B x — 2/3 2 + /3 3 . Doing this and calling the last three quantities b, c, d — -a change implying the substitution ft = b, /3 2 = b-c, f$ 3 = b - 2c + d , — we can by using the equations of condition obtain the requisite expressions for ^ , At , /3 5 , . • • , 7i , 7 2 j • ■ • • in terms of b, c, d. The theorem to which this course ultimately leads is — The continuant A a 2(»-l))8 1 . "7! A 2 (ra - 2)/? 2 . . (n + l)y 2 A, (n-S)ft (» + 2)y 3 A, is resolvable into linear factors if ft m = b - (m - \)r + x ■ od y,„ = b + mc - •2(2m- 1) (m+ \)m E , 2(2m-l) A,„ = a-2(m-l) 2 c + — — ^_ • {2?»(m-2) + 3ra}-5tf , (VII) (2m-3)(2m-l)' ^e s th factor being a + 2(u-2s + l)b - 2(«-l)(2n-2a+l)e + (n - s + 2)(s - l)-5d For example, when n = 4 we have a 2-3/; i(b + a-5d) a-2c + 20d 2(b-c) 5(6 + 2c -5^) a-8c + 24d b-2c + d 6(b + 3c-M) a-l8c + 36d = (a + 66)(a + 2b - 10c + 20d)(a -26-1 2c + 3CW)(a - 66 - 6c + 30d) . On putting d = the /3's and 7's form equidifferent progressions, and the theorem degenerates into that of § 2. (10) Out of this effort to obtain greater generality an unexpected and curious result arises ; for, whereas at first sight both members of the identity are functions of the four variables a , b , c , d , it is found on careful examination that the right-hand side is expressible as a function of one less. In fact it is readily verified that the s th factor given above can also be written in the form {a + 2(n-l)b} - 2(s-l){26-3c} - (n-s + 2)(s-l){ic-5d} so that the factorial expression for the continuant of the n th order, besides being {a + 2(n-l)b\ ■ {a + 2(n-3)b-2(2n-3)c + l-n-5d} ■ {a + 2(n - 5)6 - 4(2?; - 5)c + 2-(ra - l)-5d} ■ {a 4 2(» - 7)6 - 6(2» - 7)c + 3-(w - 2)-5d} {a - 2(?i - 1)6 - (2n - 2)- 1 -c + (» - \)-2 5d}, 350 DR THOMAS MUIR ON is also X-{X 2Y-l-n-Z} • {X-4Y-2-(w- 1)-Z} • {X-6Y-3-(w-2)Z} ■ {X-(2w-2)Y-(w-l>2Z} where X = a+2(»i-,l)6, Y = 26-3c, Z = 4c-5d. And as a , 6 , c , d cannot conversely be expressed in terms of X, Y, Z alone, the left-hand member of the identity, that is, the continuant, can only be made to appear as a function of X, Y, Z and one of the four a , b , c , d. Consequently, supposing this to be done, and thereafter all terms involving X, Y, Z deleted, we shall obtain a continuant which not only vanishes but which can be viewed as having n vanishing factors. (11) To obtain this nil-factor continuant there is, however, a better method. For, as it is the special case where X , Y , Z vanish, it must be obtainable by putting 4c = bd , or, therefore, by putting 2b = Sc=—d, 4 2(« -!)/>= -(«- 1) 15 d = 15 Doing this we find from § 9 c = - e, 6 ' h = e, a = - (7i - l)2e . /-m 1 2m -l 6 ' y m 1 2m- l 6 ' A 2{n + 2(ro-l)»-l} . -"■m (2m-3)(2m-l) and have the following theorem : — The value of any continuant of the form spoken of in § 9 is not altered by adding to its matrix the matrix of the continuant 2 ■(re-l)e 2-(ra-l)e -we ^g(n+l)fl (ft -2)1* 1 ( w + 1 )4 7?U» + 7 ) r ' (w-3)le -(w + 2)lfl ~(n+\7)e 5 n-< (VIII) CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. 351 (12) There is a special case of the theorem (VII) in § 9 which deserves particular attention, viz. the case where o- vanishes, i.e. where 5d = 2b + c. In this case the .s ,th factor = {a + 2(n-l)b} - 2(s-l){2&-3c} + (n -s + 2)(s - l){2b- 3c} = {a + 2(n-l)b} + (n-s)(s-l)(26-3c); and the (n — s + l) fch factor = {a+2(n-l)b} - 2(n-s){2b-3c} + (s + l)(n-«){26 - 3c} = {d + 2(»-l)6} + (w-s)(s-l)(26-3c). This means that tuAew 5d = 2b + c in the continuant of § 9 £Ae s th factor from the beginning is the same as the s th factor from the end, and consequently that an even- ordered continuant of this hind is a square. . . . . . (IX) (13) The question of the generalisation of the theorem of § 3 may be investigated in a manner perfectly similar to that followed in the preceding paragraphs with regard to the theorem of § 2. The essential point of difference is to be found in the new set of column-multipliers, which are now all of the form C r+2s ,._ x instead of -fj r +i-i,i»-\' It will suffice merely to enunciate the results. The first is — If the continuant | a (ra-l)ft • (n + 2)y l a + p (n-2)fi 2 . .... (w + 3)y 2 a + q (»-3)0j .... (n + 4)y 4 a+r .... be resolvable into linear factors by means of the set of column-multipliers 1 51 S 4- K C 1, 4, 10, 20, Cr+2,3 1, 6, 21, C r+4 , 5 1 > 8 > C r+6 , £/jeri (l) every four consecutive /3'.s are connected by a linear relation, viz. JKft -2ft + 00= 9(&-2& + /3 4 ), 5(/3 2 -2p 1 3 + &) = ll(/3 3 -2# 4 + /? 5 ), £/ms making all the /3's expressible in terms of any three ; (2) all the y's are expressible in terms of the same three fi's because of the fact that for all values of m (3 m + y m = — 9j8j + 25/3 2 — 14/8 3 ; and (3) p, q, r, . . . are also so expressible because of the mode of removing the first factor from the continuant. ..... (X) It should be noted that the linear relation connecting the first four consecutive |3's is that which in § 8 connects the second four, — that, in fact, the relation here is (2r + l)\/3 r - 2/3 r+] + /3 r+2 } = (2r + 7){/?, +1 - 2ft +2 + ft r+3j whereas in § 8 it is (2r - 1){/>V - 2/3,. +1 + fS r+2 } = (2r + 5){/3 r+1 - 2/3 r+2 + /3 r+3 } • TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 13). 52 352 DR THOMAS MUIR ON Further, there is a similar difference in the expressions for o- : here the expression is -9ft + 25/3 2 -14/3 3 , whereas in § 8 when given in terms of /3 2 , /3 g , /3 4 it is -9ft + 25j8 B -H£ 4 . The second theorem is — The continuant A, (M-l)^ (» + 2)yi A 2 (»-2)ftj («+3)y 2 A 3 (to — 3)jQ 3 (m + 4) 73 A 4 is resolvable into linear factors if the s th factor being (m - l)(m -2) „ j A.- & -<»-i)« + S<2,£t) 7d ' ym=&+(m + 2)c _ (^3)(^2) , 7ri; ' v 7 2(2m +1) A w = a - 2(m 2 - 1 )e + ^4^(2m 2 + 2 + 5n) • Id , a + 2(7i - 2s +\)b- 2(« - l)(2n - 2s + 3)c + («-s + 4)(.s-l)- 7d, or {a + 2(«-l)6}-2(s-l){26-5c}-(n-s + 4)(s-l){4c-7^}. . . . (XI) An immediate deduction from this is that when 14d = 2b + 3c the 8 th factor is a + 2(n - 1)6 + }(» - »)(a - 1)(26 - 5c) and is the same as the 8 th factor from the end, so that when in addition n is even the continuant is a square. ....... (XII) The third theorem is — The value of any continuant of the form referred to in § 3 is not altered by adding to its matrix the matrix of the continuant 2 1 - ^(n + 2)e 3-5 (m - 7)e 1 (»- 2)e - J(«+3)e _ jL(ra-17)e i(n-3)« 5 5-7 7 1 'n + i)e ■■9 (»- 31)e . XIII) (14) The corresponding theorems for the set of column-multipliers 1; 1) 1; 1 > 1 j J ^i-l, 1, 2, 3, 4, , C,.,, 1.4 CU 9 will be found in the Trans. S. Afr. Philos. Soc. referred to above. CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. 353 When the first theorem there given is attempted to be generalised in the manner employed in the present paper the following is the result : — The continuant of the u th order ivhose main diagonal is a, a + (b-c)-(n-2)y, a + 2(b- c) - 2(w - 4)y , a + 3(&-c) - 3(n- b)y , .... and ivhose minor diagonals are (n-\)b, (n-2)(b + y), („-3)(6 + 2y), . . . . 2{c-y), 3(c-2y), .... is equal to the product of the n factors {a + (n-l)b\ ■ {a + (n-2)b-c+l ■ (n-2)y} ■ {a + (n - 3)6 - 2c + 2 • (n - 3)y } • { a - (n - 1 )c) (XIV) It is seen to degenerate into the original theorem when y is put equal to 0. If, how- ever, we write X in it for the half-sum of the first and last factors, and Y for b + c , the factors may be written X + i(n-l)T, X + J(ra-3)Y+I -(n-2)y, X + |(w-5)Y + 2-(«-3) r , X-l(«-l)Y; thus showing that four variables are not necessary for the expression of the identity. An easier way of reaching the same result is to put a = a-(n- 1)£ , b = fS + $, c=P-£, when it will be found that .£ appears in the continuant but not in its factors ; and when there are consequently obtained at one and the same time the case of theorem (XIV) where b = c, and an expression for the corresponding nil-factor continuant. (15) We have thus in all at present three sets of column-multipliers, each of which has associated with it a linearly resolvable continuant of the form 7i Pi a + p y-2 a + q 7s a + r Other sets will doubtless be discovered, as the only difficulty is the devising of a set which will not lead to unreasonably complicated expressions for the elements of the continuant. In all cases if fc 1 ) fe 2 ' f S ' ' • " • ' =«-! 354 DR THOMAS MULR ON be the diagonal adjacent to the diagonal of units in the set, — that is to say, if the set be 1 , £ , 1 , l 2 , — and if we denote the elements of the resolvable continuant by their place-names (1, 1) , (1,2), . . . , then the factors of the continuant are {(1,1) +£■(!, 2)} ■ {(2 s 2)-£.(l,2) + ! 2 .(2,3)} .{(3,3)-| 2 .(2,3) + 4.(3,4)} {(n.fO-^-fn-l.n)} (XV) A scrutiny of the procedure connected with the removal of any factor makes this evident. For, firstly, when s — 1 factors have been removed, the residual determinant has for its first column the line of column-multipliers last used, viz. 1, f_i, ; secondly, this determinant when reduced to the next lowest order has (s,s)-L_ 1 -(s-l,s) for the first element of its diagonal ; thirdly, the employment of the next line of column-multipliers, viz. changes the said element into (s , s) - 4-1 -(8-1,8) + Us ,s + l); and this, in virtue of the character of the process, is the next factor ready for removal. It may be noted in corroboration of § 6 that the sum of the factors thus expressed is (l,l) + (2,2) + (S,3)+ . . . + (n,n). (10) Observing from the foregoing that (n ,n) - i n -i(n - 1 , «) is the last factor, we have suggested to ourselves the obtaining of the factors in the reverse order by the use of a set of roio-multipliers, the first operation being row, - £„_! • row,,.! + . . . . An interesting result is thus reached, viz., that corresponding to each set of column- multipliers for the resolution of a continuant there is an equally effective set of row- multipliers. Thus returning to the continuant of § 2 and performing the operation row 5 - 8 row 4 + 28 row 3 - 56 row 2 + 35 tow 1 we find we can remove the factor a — 8b — 8c, and write its cofactor in the form a 2-4:1 5(6 + c) a -2c 3(6 -c) 6(6 + 2c) rt-8r: 2(6 - 2c) 7(6 + 3c) a -18c 1(6 -3c) 35 -56 28 -8 1 CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. 355 a 5(b + c) and thence in the form 2-46 a - 2c 3(6 - c) 6(6+ 2c) a -8c 2(6 -2c) -35(6 -3c) 56(6 -3c) -21(6 -5c) a + 86 -42c Similarly the operation row 4 - 6 row 3 + 15ro\v 2 — lOroWj now enables us to remove the factor a — 4b— 1 8c ; and the operations row 3 — 4 row 2 + 3 rowj , ro\v 2 - 1 rowj the remaining factors. The set of column-multipliers 1, 1, 1, 1, 1 1, 4, 9, 16 1, 6, 20 1, 8 1 is thus equivalent to the set of row-multipliers 1 , - 8 , 28 , - 56 , 35 1, -6, 15,-10 1, -4, 3 1, -1 1 . (17) The general result is that the table of row-multipliers suitable for the resolution of the continuant of § 2 is 1 ) — Cjn-2, l , ^2(1-2, 2 J — ^--'n-2, 3 I 1 ) — ^"2n—i, 1 ) C^i-4, 2 ) > ( — )" f C. 2 , t _2, n_l (XVI) Similarly it is found after a little investigation that the table of row-multipliers suitable for the resolution of the continuant of § 3 is 2 2 2 1 , - y(» - l)O lB _ li o , y(ra - 2)C,„_ 1 , 1 , - — (n - 3)C 2n _ 1 ,, 2 , 1 , -hn- 2)C 2B _ 3 . 01 , |>- 3 ) C ^> . (XVII) the general form of the multiplier being 2^C 2r+liS . Lastly, the table of row-multipliers suitable for the resolution of the continuant oj § 13 is 1 > "Ui-i,n "ji-i,!j — ^h-1,3) ~1 1 , -C_*i. C„_, a , - • (XVIII) -that is to say, may be got by a rearrangement of the column-multipliers : for 356 DR THOMAS MTJIR ON example, in the case of the 5 th order the equivalent tables of column-multipliers and row- multipliers are 1,1,1,1,1 1,-4, 6,-4, 1 1,2,3,4 1,-3, 3,-1 1,3,6 1,-2,1 1, 4 1, -1 1, 1. (18) There falls now to be noted a set of theorems regarding resolvable continuants of a totally different form but connected with and derivable from those of §§ 2, 3, 13. If in any one of these latter theorems we put for the element in the place (l, 1) , the continuant is expressible as the negative product of the elements in the places (1, 2), (2, 1), and a continuant of the lower order n — 2: further, one of the said elements is contained in the first factor of the original continuant and the other in the last factor : in this way, therefore, the resolution of the new continuant of order n — 2 is secured. Thus, taking the five-line continuant dealt with in § 2 and putting a = we obtain ■4-56(6 -cj 8c 2(6 + 2c) 7(6 -3c) 18c 1.(6 + 8c) 8(6 -4c) 32c = 86(46 + 14c)(20c)( - 4b + 18c)( - 86 + 8c) , and therefore 8e 2(6 + 2c) 7(6 -3c) 18c l-(6 + 3c) 8(6 - 4c) 32c The general theorems thus obtained are (46 + 14c) -20c -(-46 + 18c). A 1 (»-l)/8j (?j + 4) 7l A 2 (n + 5)y 2 (n - 2)& = 2- n 1 ■ I 2(n-l)6-2(2» + l)c+l.(n + 2)-5d| ■it + 21 j {2(n - 3)6 - 4(2ra - l)c + 2-(» + l)-5<*} {2(n - 4)6 - 6(2n - 3)c + 3-n-5d} (XIX) if « z. / , i\ , !«(ffl+l) c , i , , , n\ (m+ 2)(m + 3) K , K = b - {m + 1), + g^L—JW , y m = I + (« + 2 )C - 2(2 J +3) 5,/ . and (»-!)& A,„ = (m + 1) 2 (rc+6)y A, (n + 7)y. (n-2)& A. _ 9 , 2m(m + 2) + 6 + 3?* ,- , \ (2m+l)(2m + 3) ' J ' = 4- ra+ - 1 • | 2(»-l)6-2(2rc + 3)c + l-(re + 4).7<2 {2(n - 3)6 - 4(2w + l)c + 2-(ra + 8).7d} {2(« - 5)6 - 6(2n - l)c + 3-(ra + 2)-7d} / (XX) if r-i i i , i\ , rn(v) + l)- 7 . .. (w + 4)(m + 5) „ , p,„ = w — (vm + l)c + — i ' la , y.„, = 6 + (?». + 4)c - ^ ; — ^ — — — '- til , H V ' 2 2m + 5) ' 7 '" K ' 2(2m + 5) and A„, = (to+1)(to + 3) -2c + 2wi 2 + 8to + 10 + 5m (2m + 3)(2n» + 5) 7-/ CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. 357 2(b- c - ray + 2y) In - 1 )(b + 2y) 3(c-2z) (36-c-ray+4y) (ra-2)(6+3y) 4(c-3y) 4(6-c-ray + 6y) (XXI) = (ra+l).{ra( i 8 + y)^c}{(ra-l)(6 + 2y)-2c}{(ra-2)( / Q+3y)-3c} Of these three results it would be interesting to obtain independent proofs. (19) In a paper of Heine's on Lame's functions (Crelle's Journ., lvi. pp. 77-99) there occurs the continued fraction 'V-'i 2 C ft C<,C S Z — C-, — r, where the t's are m in number with the values \m(m+l) , {(m - l)(m + 2) , J(m-2)(m + 3), . . . . , \-\-2m, the fractional factor being £ in every case except the first. From extraneous considera- tions the value of the continued fraction was given z( 2-2 2 )(z-4 2 ) . . . for m even and (2 - to 2 ) -( Z - 22 ).,,.( Z -^) f ° rm0dd but the author added, " Einen directen Beweis fur diese Summirung des Kettenbruches habe ich noch nicht aufgefunden." This, however, is readily obtained by writing the value of the continued fraction as the quotient of two determinants, viz. z- c, h '■'„ "0 C-i Z Cj — C.) f- 2 C 3 Z ~ C 3 ~ r 4 Z - C-, - C and then using two of the foregoing theorems. Thus, taking the case where m = 6, and where therefore the c's are 21, 10, 9, 7i, 5J, 3, we have to evaluate the quotient 2-21 21 10 z-19 71 2-13 51 2- 19 9 -1 3 2-3 7i 2 - 13 5-| I . . 3 2-3 the dividend of which, changed into 2-21 6-3i 4(3^-1) z-21 + 2-1 2 2.(31+1) 5(31-2) z-21 + 2-2 2 l-(3J + 2) 6(3£-3) 2-21 + 2-3 2 358 DR THOMAS MUIR ON CONTINUANTS RESOLVABLE INTO LINEAR FACTORS. is seen to be one of the simplest cases of the determinant of § 2, and thus to have for its value g(a-4)(2-16)(z-36>. In like manner the divisor may be written z-19 2-4J 3(4£-2) 2-19 + 6 1(4£-1) 4(4£-3) z-19 + 16 and is then recognised to be a very special case of the continuant of § 3, and therefore to be equal to (*-l)(«-9)(«-25). The continued fraction in question is consequently equal to 2(3-4)(2-16)(z-36) (2-l)(2-9)(2-25) ' ( 359 ) XIV. — The Igneous Geology of the Bathgate and Linlithgow Hills. By J. D. Falconer, M.A., B.Sc. Communicated by Professor James Geikie, LL.D., D.C.L., F.R.S. (With a Map.) (Read December 5, 1904. MS. received same date. Issued separately June 9, 1905.) CONTENTS. PAGE PAGE Previous Literature 359 The Fourth Volcanic Zone, or The Hilderston and Introduction 359 Hiltly Lavas 362 The Houston Coal 360 j The Index Limestone and the Bo'ness Lavas . . 363 The First Volcanic Zone, or The Brox Burn Ash . 360 The Fifth Volcanic Zone, or The Kipps and Bishop- The Second Volcanic Zone, or The Longmuir and brae Lavas 364 Riccarton Lavas 361 The Dykeneuk and Castlecary Limestones . . 365 The Third Volcanic Zone, or The Kirkton and Hill- ' The Volcanic Necks 365 house Lavas 362 The Intrusive Rocks 365 The Hurlet Limestone 362 General Results 366 Previous Literature. Geological Survey, sheet 32, 1859 ; revised ed. 1892. sheet 31, 1875. Geology of the Neighbourhood of Edinburgh ; Memoir of sheet 32, 1861 ; Memoir of sheet 31, 1879. H. M. Cadell, " The Geology of the Oil Shalefields of the Lothians," Trans. Ed. Geol. Soc, 1901. "The Volcanic Rocks of Bo'ness," Trans. Ed. Geol. Soc, 1880-1. in Trans, of Inst, of Mining Engineers, Glasgow Meeting, 1901. Introduction. The Bathgate and Linlithgow Hills occupy a well-defined belt of rising ground stretching S.S.W.-N.N.E. from Bathgate to Bo'ness, and included in the marginal portions of sheets 31 and 32 of the 1-in. maps of the Ordnance and Geological Surveys. Throughout the range the steeper slopes face the west, while the eastern flanks are deeply buried in drift. The glaciated contours are well seen from the east, and the sky-line is in several places deeply indented by glacial grooves. The physical geography is throughout intimately dependent upon the geological structure, but the latter is simplicity itself when contrasted with the complicated structure of the shale- fields to the east. Alternating zones of volcanic and sedimentary rocks strike parallel with the direction of elongation of the range from Bathgate to Bonnytoun Hill, and these are cut by a later connected series of dykes and sills of intrusive igneous rock. The latter, as a rule, are more resistant than the lavas, and form the more prominent features of the landscape, while the sedimentary intercalations may frequently be traced, even where no rock is visible, by the trough-like depressions which have been hollowed out of them between the zones of lava. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 14). 53 360 MR J. D. FALCONER ON THE As convenient boundary -lines for the study of these hills, I have chosen as lower limit the final western outcrop of the Houston Coal from Drumcross to Champfleurie, and thence a line drawn directly north to the shore of the Firth at Stacks, and as upper limit the outcrop of the Castlecary Limestone. The accompanying map represents on a reduced scale the 6-in. sheets ix. N.E., v. S.E., v. N.E, i. S.E., i. N.E. of Linlithgowshire, with the eastern halves of ix. N.W., v. S.W., v. N.W., i. S.W., i. N.W. of the same county. I have thought it unnecessary to continue the map to the south of the Bathgate Railway, beyond which the volcanic series rapidly runs out, while the country is so much covered with drift that little rock is visible. The geology below the Houston Coal and above the Castlecary Limestone is simply sketched in from the Survey sheets, and has not been personally verified in detail. The Houston Coal. The outcrop of the Houston Coal, crossed by numerous dip faults, has been well proved from Deans to Drumcrosshall. Thence it strikes north by west to the neighbour- hood of Blackcraig and West Binny, having been worked many years ago at both of these places. In the vicinity of Ochiltree Mill the outcrop is uncertain, the ground being much faulted and pierced by many intrusions. North of Peace Knowe, however, it reappears and strikes north by west to the Haugh Burn fault, by which it is shifted to the east beyond the limits of the present map. Between the Houston Coal and the first volcanic zone come the Houston shales and marls, with thin sandstones and occasional beds of ash and agglomerate. Good sections are found in the Mains Burn and the Brox Burn and its tributaries. The First Volcanic Zone, or The Brox Burn Ash. This zone can be traced, with interruptions, from Drumcrosshall to the Haugh Burn. It consists throughout of stratified volcanic ash, varying much in texture and usually green in colour, but weathering yellow or brown at Chapelhill and Bankhead. Interbedded lava is nowhere found in this zone, although in places the compact ash weathers spheroidally, and presents a deceptive resemblance to a decomposing crystalline igneous rock. Characteristic sections are found in the Brox Burn and its tributary to the south. On account of the absence of exposures, this zone cannot be traced to the east of the Longmuir plantation. It reappears, however, on the same horizon on the Riccarton road, immediately to the north. Between this ash and the second volcanic zone there appear some thinly bedded sandstones and shales, frequently ashy themselves, and interstratified with thin bands of ash. These are well seen in the Brox Burn and in a streamlet in the northern angle of the Longmuir plantation. IGNEOUS GEOLOGY OF THE BATHGATE AND LINLITHGOW HILLS. 361 The Second Volcanic Zone, or TJie Longmuir and Riccarton Lavas. This zone is taken to include two apparently distinct groups of lavas. The lower or Longmuir group extends from a point a little to the east of Broomyknowes, through the Longmuir and Balditop plantations, to Drumcross. The upper or Riccarton group is of greater thickness but of less extent, stretching only from the Riccarton Burn to the Rigghead plantation, a little to the south of Tartraven. The two groups are separated by a series of sandstones, shales, and thin ashbeds, well seen in the streamlets and quarries on the northern slope of the Riccarton Hills. Petro- graphically, the Longmuir lavas are throughout fine-grained olivine-basalts, while the Riccarton lavas can be separated into a lower zone of coarse-grained olivine-dolerites # extending the full length of the group, and an upper zone of olivine-basalts stretching from the Riccarton Burn to North Mains. This subdivision of the zone is everywhere apparent to the north of the Mains Burn, but to the south so little rock is exposed that the lines are to some extent hypothetical. The sedimentary intercalation between the two groups of lavas cannot be traced by means of actual exposures of sandstones and shales. In consequence, more than usual reliance must be placed upon the prolongation to the south of the petrographical variations established above. The coarse-grained olivine-dolerites of the lower zone of the Riccarton lavas are nowhere found to the south of the Rio-ahead plantation, beyond which the exposures are all of olivine-basalts similar to the Long- muir lavas to the north. The sedimentary band, apparently reduced in thickness, is therefore drawn, as nearly as possible, between the basalts and the dolerites. Marginal sedimentary intercalations are probably numerous throughout this zone. Two such are shown on the map, one of shales and sandstones found in drains on the farms of Drumcross and Quarter, the other of sandstones, shales, fine-grained green ash, and agglomerate, exposed in the Brox Burn at the Balditop plantation. The sections exposed during the construction of the Bangour reservoir indicated considerable dis- turbance of the strata at that point. Another thin bed of sandstone, interstratified with the lavas, may be seen immediately to the east of the neck in the Riccarton Hills. This second volcanic zone cannot be definitely traced to the south of the Galabraes fault, although the section at Starlaw indicates the occurrence of scattered lava-flows. North of Broomyknowes also the lavas rapidly run out, and their place is taken by interst ratifications of sandstone, shales, and thin ashbeds, with at least one thin band of shelly limestone exposed in the burn below Riccarton Mill. Between the second and third volcanic zones there comes another sedimentary intercalation, represented by shales and thin ashbeds at Craigs, limestone and shale at Tartraven, shales and ash in the Riccarton Burn west of Beecraigs, sandstone, shale, and limestone at Whitebaulks, sandstone at Hillhouse, sandstone and limestone at * The terras "basalt" and "dolerite" are used throughout to denote macroscopic distinctions, "basalt" implying a fine-grained, compact, and usually porphyritic rock, " dolerite " a coarse-grained rock, not evidently porphyritic. 362 MR J. D. FALCONER ON THE Carsie Hill, east of Cauldhame, and at Peat Hill, on the north side of the Haugh Burn fault. The Third Volcanic Zone, or The Kirkton and Hillhouse Lavas. This zone includes the Kirkton, Tartraven, and Hillhouse lavas. A very charac- teristic ash, with black matrix and yellow lapilli, lies in several places at the base of this zone. It is well seen at Whitelaw, Craigs, Whitebaulks, and Hillhouse. The lavas of this zone can be studied with ease in numerous exposures from Kirkton Mains and Boghall to The Knock. Porphyritic olivine-basalts predominate, but a thin band of dolerites strikes N. W. from the Raven Craig. The limestones and accom- panying shales and ash of the east and west Kirkton quarries, so well described in the Survey Memoir accompanying sheet 32, occur as isolated lenticular patches between successive lava-flows, and evidently occupy a much higher horizon than the Tartraven Limestone. A similar intercalation of sandstone may be seen on the eastern slope of the Knock Hill. From the Knock to Tartraven little rock is visible, but numerous exposures are found in the Tartraven Hills where the road cuts through a series of dark-blue lustrous limburgitic olivine-basalts. The lavas of this zone probably run out to the north of the Mains Burn, for in a streamlet to the east of Balvormie the only representative of the zone is the basal ash noted above. This apparently swells out by Whitebaulks to Hillhouse, where it dips below a group of coarse-grained olivine-dolerites, which, after suffering displacement by the Haugh Burn fault, runs out to the north of Parkly Place. The Hurlet Limestone. This well-marked horizon can be traced from Glenbare quarry, east of Bathgate, to the North Mine quarry on the Tartraven road. For a mile to the north of this point the outcrop is conjectural, no trace of the limestone being found at the surface. It reappears, however, in characteristic sections in the Hillhouse and Hiltly quarries. North of Hiltly the outcrop must be shifted to the east by the Haugh Burn fault, and probably strikes north from the vicinity of Parkly Place to Linlithgow Poorhouse, and thence north-west to the shore at Stacks. The Hurlet Limestone is throughout associated with sandstones, shales, and thin ashbeds ; and detailed descriptions of sections, formerly better visible than now, may be found in the Survey Memoirs and Mr Cadell's papers. To the east of Linlithgow no volcanic rocks are found below the Hurlet Limestone, within the limits of the present map, and, other than the thick sandstone formerly quarried at Kingscavil, little rock of any kind is visible. The Fourth Volcanic Zone, or The Hilderston and Hiltly Lavas. This zone reaches its greatest thickness immediately to the south of Linlithgow, but even here the apparent thickness is greater than the actual thickness, on account of IGNEOUS GEOLOGY OF THE BATHGATE AND LINLITHGOW HILLS. 363 the effect of the Haugh Burn fault. Petrographically, the zone is composed of a number of alternating bands of olivine-basalts and olivine-dolerites. Transverse traverses in the neighbourhood of Clarendon, Hiltly, Wairdlaw, or Hilderston readily reveal this structure. As a rule, however, these bands cannot be traced far in a north and south direction. A stratified intercalation of ash and ashy shales is found in Preston Glen, and another of limestone and shales at Wairdlaw. The basalt overlying this limestone is noteworthy, both for its platy jointing and for the fact that it is the only lava throughout the whole volcanic series which contains phenocrysts of felspar in any abundance. The same rock can be traced on the south side of the valley at Wairdlaw, but the limestone below is nowhere visible. A line of springs behind Craigmailing probably marks the prolongation to the south of this sedimentary intercalation, and indications of its presence are also found on the eastern slope of Cathlaw Hill. Towards the south interbedded sediments probably become more abundant. Two bands are exposed in the Mavis Glen, and these can be traced for a considerable distance to the north and to the south by means of the shallow depressions between the lavas to which they give rise. The lavas of this zone cannot be traced to the north of Linlithgow Loch. In all probability they rapidly run out, and their place appears to be taken in part by a thick bed of volcanic ash found in a bore in Bonnytoun farm. Farther north the sandstones, shales, and thin limestones of Carriden probably occupy approximately the same horizon. The Index Limestone and the Boness Lavas. Between the fourth and fifth volcanic zones there occurs in the Bathgate Hills an important belt of sedimentary rock, which includes the lower Bathgate coals and the Index limestone. It retains a fairly uniform thickness and character from Bathgate to Kipps. North of Kipps, however, the thickness gradually increases, and volcanic material becomes mingled with the sedimentary. On the eastern slope of Cockleroy a bed of ash appears not far below the probable position of the Index limestone, and at Kettlestoun fine-grained volcanic mudstones, passing into ashy sandstones and shales, are found overlying the lavas of the fourth volcanic zone. Between Kettlestoun and Linlithgow Bridge the only exposure is in the river Avon at the railway viaduct, where a very vesicular basalt lies a few feet below the Index limestone. The journals of bores quoted by Mr Cadell seem to indicate that a considerable proportion of the rock below the glacial gravels of this district is of volcanic origin. This change in the character of the strata makes it very doubtful whether the Kipps coals and the Index limestone are continued across Cockleroy to Kettlestoun. It is quite possible, however, that the coals do exist, but almost certainly in an attenuated form, of no commercial value, and much destroyed by intrusive rock. North of the Edinburgh and Glasgow Railway the Index limestone is repeatedly exposed in the river Avon, while the sedimentary zone, as a whole, opens out rapidly 364 MR J. D. FALCONER ON THE to include the Bo'ness coalfield and its intercalated volcanic rocks. In one sense, there- fore, since mere thickness of strata is in this case no index to rate of deposition, the whole of the Bo'ness coalfield may be considered the equivalent of the lower Bathgate coalfield. On the other hand, from the position of the igneous material where it first appears at Coekleroy and Linlithgow Bridge, the lower Bo'ness coals might possibly be considered more nearly the equivalent of the lower Bathgate coals in point of time. Further, the lavas of the Bo'ness coalfield apparently form a group by themselves entirely distinct from the volanic zones of the Bathgate Hills to the south. It is highly probable, as Mr Cadell has suggested, that the conditions of sedimentation were entirely different on either side of a volcanic orifice somewhere in the vicinity of Little Mill. It is unnecessary to describe in detail the stratigraphy of the Bo'ness coalfield. That has been admirably done already by Mr Cadell, and the Little Mill district alone still remains more or less of a puzzle. Petrographically, the lavas of Bonnytoun Hill, south of the Roman road, can be readily subdivided into three zones — a lower zone of coarse-grained olivine-dolerites between the Red Coal and the Wandering Coal ; a middle zone of porphyritic olivine basalts between the Wandering Coal and the Western Main Coal ; and an upper zone of coarse-grained dolerites between the Western Main Coal and the Muirhouse coals. The middle zone alone can be traced below the ash of Little Mill to Pepper Hill and Linlithgow Bridge. To the north the lower zone can be traced continuously to the Bonhead fault, but the two upper zones, north of the Roman road, apparently pass into finer-grained doleritic basalts, which persist throughout the remainder of the coalfield. The rock exposed above Bonsyde is similar to the lavas of the middle zone to the west, and is probably a displaced portion of the other lavas of the hill. The Fifth Volcanic Zone, or The Kipps and Bishopbrae Lavas. This zone lies a short distance above the Index limestone, and may be traced, with interruptions, from Linlithgow to Bathgate. Between the Avon Paper Mills and the vicinity of Coekleroy, where this zone reaches its greatest thickness, the only ex- posure is found in the Cauld Burn at East Belsyde. It is highly probable, however, that this zone is continuous throughout. Petrographically, the lavas belong mostly to types of olivine-basalt, and limburgitic varieties may be studied with ease in the neighbourhood of Kipps. Coarser-grained doleritic types occur here towards the summit of the series, and appear also in the river Avon at Linlithgow. Above the lavas and below the Dykeneuk limestone evidence of the continuance of volcanic action is found in Carriber Glen, where a thick series of ashy sandstone and volcanic mudstones, in places fossiliferous, are exposed in the gorge of the river Avon. Similar ashy sandstones are found at Threegables, east of Bowden Hill ; and in a streamlet between Lochcote and Gormyre a bed of ash occurs on approximately the same horizon, overlaid by a peculiar blue mudstone, which in places much resembles a decomposed igneous rock. IGNEOUS GEOLOGY OF THE BATHGATE AND LINLITHGOW HILLS. 365 The Dykeneuk and Castlecary Limestones. The Dykeneuk limestone, though proved in many bores, is seen at the surface in three exposures only — at Dykeneuk, Woodcockclale, and Carribber. The Castlecary or Levenseat limestone is exposed at Craigenbuck, at the Birkhill viaduct, the Avon aqueduct, Carribber, Bowden Hill, and Lochcote. Both limestones are probably con- tinuous across Bowden Hill, though in places cut out by intrusive rock. South of Bishopbrae their presence beneath the surface has been repeatedly proved in bores. The strata between the two limestones consist of sandstones and shales, thickest towards the north and thinning out towards the south, with the effect of bringing the limestones closer together from Carribber southwards. No trace of volcanic activity is found above the Dykeneuk limestone. The Volcanic Necks. These are found in the eastern part only of the volcanic area. A small neck full of green ash pierces the Riccarton Hills south-east of Belcraigs. Another, full of coarse agglomerate, breaks through the stratified ash of the first volcanic zone to the west of Wester Ochiltree. A group of seven small necks is found in the neighbourhood of Hiltly and Parkly Place, some filled with coarse agglomerate, and others with fine- grained ash similar to that at the base of the third volcanic zone. The Necks of Pilgrim's Hill and Carriden are also included in the accompanying map. The Intrusive Rocks. The intrusive rocks of this district are readily separated into two groups according to their microscopical characters, — a smaller group of olivine-basalts and dolerites, and a larger group of augite diabases, with little or no olivine, but with frequently abundant hypersthene. An intersertal microlitic or micropegmatitic groundmass is usually present in the latter, in greater or less abundance. This difference in mineral composition seems to be most easily explained on the assumption that the two groups are the products of different periods of igneous activity. The smaller and more basic group might readily have been produced from the consolidation of igneous material similar to that which produced the lavas. They may therefore be regarded as more or less contemporaneous intrusions. The large group, however, is of a more acid character, and is certainly the product of a later period of igneous activity. (a) Contemporaneous intrusions. — Small intrusions of olivine-basalt are found in many of the Necks enumerated above, but, other than these, few contemporaneous intru- sions have been recognised throughout the volcanic zones. A small intrusion of olivine- basalt cuts the ash of the first volcanic zone in the Brox Burn near the Bangour reservoir. A short dyke of similar material occurs in the third volcanic zone north- 366 IGNEOUS GEOLOGY OF THE BATHGATE AND LINLITHGOW HILLS. east of Wairdlaw, and another at Haddie's Walls in the Kipps coalfield. The small dyke piercing the lava-cliff to the west of Hiltly is probably of contemporaneous origin. Much also of the intrusive rock in the vicinity of Ochiltree Mill, as well as the Walton and Carriden intrusions, may be referred, from petrographical characters, to the first period of igneous activity. (b) TJie later intrusions. — These may be subdivided into two groups : — 1. Dyke-like intrusions with a vertical or highly inclined junction — E.g. The Raven Craig, the Knock, the Witch Craig, the Wairdlaw and Cockleroy intrusions, and the E.-W. dykes. 2. Sills or laccolitic intrusions — E.g. The Kettlestoun Hills, the Belsyde Hills, and the Torphichen Hills. The similarity in petrographical type of these intrusions indicates contemporaneity of origin in late or post-carboniferous times. Strict contemporaneity, however, is not implied, the intrusions having evidently been inserted in succession during the period. Those intrusions with a N.-S. elongation are probably oldest, being cut, as in the case of the Raven Craig, by the E.-W. faults, along which dykes have usually risen. These dykes are apparently the feeders of the sills, the lowest and oldest of which are some- times covered by the feeders of the higher and younger. The upper limits of the dykes themselves were probably irregular, different portions of the same dyke rising to different levels. The upward termination of one of these dykes is particularly well seen at Broomyknowes and Belcraigs. This may explain in part the discontinuity of outcrop of some of these dykes when traced from east to west. In other cases, how- ever, as in the Parkly Craigs dyke, the different portions seem to run out, the dyke being continued on a parallel line a few yards to the north or south. General Results. 1. The lavas of the Bathgate Hills are olivine-bearing from base to summit of the series, and are pierced by a few contemporaneous intrusions of similar material. 2. The Bo'ness lavas form a group entirely distinct from the lava-zones of the Bathgate Hills. 3. The volcanic zones are crossed by a later-connected series of dykes and sills, probably of Palaeozoic age. The detailed results of the microscopical and chemical analysis of the rocks are reserved for a future paper. ( 367 ) XV. — On a New Family and Twelve New Species of Rotifera of the Order Bdelloida, collected by the Lake Survey. By James Murray. Communicated by Sir John Murray, K.C.B., F.R.S. (With Seven Plates.) (MS. received January 13, 1905. Read January 23, 1905. Issued separately March 3, 1905.) Introduction. The new species here described were found in the course of the work of the Lake Survey on Loch Ness and other Highland Lochs. Half of the number were found in lakes, though they are not exclusively lacustrine, three in ponds, two from moss growing on the shores of Loch Ness, and one in a stream running into Loch Ness. Structure. — A short account of the structure of a typical Bdelloid will be necessary, in order to render intelligible the terms used in the descriptions. A Bdelloid is a Rotifer which can creep like a leech or caterpillar. The body is segmented, and consists of head, neck, trunk, and foot. The head, neck, and foot are telescopic, and can be completely withdrawn into the trunk. The normal number of segments is sixteen, but there may be more or less, the variation being chiefly in the foot. The head consists of three segments, the neck of three, the trunk of six. It is believed that the number of segments in each of those portions of the body is invariable, but two or more segments may be so united as to be indistinguishable. The foot is more variable, the number of segments, normally four, varying from one to six. Beginning at the anterior end, the first and second segments of the head form the rostrum. The first has an inverted tip, from which rise the two rostral lamellae, numerous motile cilia, and sometimes larger tactile setae. The third segment is the oral, and bears the mouth, and the corona when present. The first cervical bears the antenna, and frequently a number of prominences. The second and third cervical have no appendages. The first, second, third, and fourth segments of the trunk are called the central, and form the broadest part of the body. The next two segments of the trunk, the pre-anal and the anal, together form the rump, which is generally clearly marked off both from the central part of the trunk and from the foot. At the end of the anal segment is the anus. The segments beyond the anus constitute the foot. The first joint of the foot commonly has the skin on the dorsal surface thickened, and often bears a rounded boss or other processes. The penultimate joint bears the spurs. The last joint bears the toes, or the perforate disc which takes their place. The segmentation is superficial, and affects only the skin. When the animal is fully extended, the various organs usually occupy definite segments, though the arrangement is not invariable. The brain, generally somewhat triangular, occupies the second cervical, but when large may extend TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO 15). 54 368 MR JAMES MURRAY ON A NEW FAMILY through all three cervical segments. The eyes, when present, are in the rostrum, near its tip, or on the back of the brain. The mastax, containing the jaws with their teeth, is in the third cervical or first central. The stomach extends through all four central segments, which also contain the ovaries and the eggs or young. The pre-anal contains the intestine, the anal the contractile cloaca. The foot-glands occupy all the joints of the foot, and may extend into the anal, or even into the pre-anal. The toes are three or four in number, or they are united to form a disc, which is perforate with pores for the passage of the mucus. The corona of the Philodinadae consists mainly of two nearly circular discs, borne on pedicels. The pHncipal wreath borders the discs. The secondary wreath runs round the bases of the pedicels from back to front, and merges in the cilia of the mouth. Near the centre of each disc is in many species a seta, or pencil of setae, or several short motile cilia, which usually rise from a small papilla or a larger process. The lower lip is the central portion of the under side of the mouth, and is shaped like a V. The upper lip is the space between the bases of the pedicels and the front of the rostrum. Its form is very characteristic for each species. Its most important structures are two folds of skin, which continue those prominences at the sides of the mouth known as the collar. These folds run round the bases of the pedicels, close to the secondary wreath, and may meet in the middle line just in front of the rostrum, or may terminate at some distance apart, in processes of various form. In the middle line, between the pedicels, is often found a peg-like process, known as a ligule. The water-vascular system, with its vibratile tags, usually about six pairs, is difficult to observe. The number of pairs of tags seen is always noted, though this may not be the full number present. The skin may be smooth and hyaline, stippled with pellucid dots, viscous, papillose, or variously warted or spiny. The back and sides of the trunk are longitudinally plicate. The ventral side is obscurely transversely plicate. In a few species the ventral transverse folds are numerous and deep. Habits. — The great majority of the known species are free and independent animals. None are truly pelagic. Even those which are in the habit of swimming only do so for short distances, and in the shelter afforded by mosses and other water plants. Those which do not swim creep in caterpillar fashion on the plants among which they live or on the mud of ponds. When feeding they anchor themselves by the foot. The Adinetadse and some Callidince can also glide forward by the action of certain cilia — in the Adinetadse those of the corona, in the Callidince those of the rostrum. Several species secrete protective cases ; others accumulate irregular tubes of d6bris. Parasitism. — A number of species are ectoparasites upon other animals. None are internal parasites or feed upon their host. They are commensals or messmates. They attach themselves by the foot to some crustacean, insect larva, or other animal. They seem to desire from their host only protection, conveyance from place to place, possibly a share of food. Asellus is a favourite host, and often carries several species together. All the Bdelloids known to me which have taken to the parasitic mode of life are large AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 369 animals. They are distinguished by bulk of body, long and powerful foot, large strong spurs, and usually ample corona. The life agrees with them. Signs of degeneration are not lacking, however. Their affinities are with genera which normally possess eyes, but most of these parasites are blind, and have smaller brains than are possessed by free- living species of the same genera. Symbiosis. — A number of Bdelloids have the foot of a peculiar type, in which there are no distinct toes, their place being taken by a disc, which is perforated by numerous pores for the passage of the mucus. It has been suggested that the species having this kind of foot live in symbiotic relationship with certain Hepatics, such as Frullania, which have some of their leaves in the form of little pitchers. The suggestion is plaus- ible, inasmuch as such Hepatics are seldom found without the pitchers occupied by the Rotifers. There is, however, something to be said against the belief that the relation is one of symbiosis. There is no evidence of any advantage to the plants, though it has been guessed that the animals are in some way beneficial to them, and that to obtain this benefit the pitchers have been developed as an attraction to Rotifers. It is more probable that the pitchers of the Hepatics serve the sole purpose of retaining moisture, and that many species of Rotifers have found and taken advantage of those little reservoirs. In that case there is no symbiosis, only a mild form of parasitism. Bdelloids having the discoid foot are not only found on Hepatics. They abound in many other situations. Nor are they the only Bdelloids which frequent Frullania cups. Many species with the ordinary foot are commonly found there. It might be supposed that a Bdelloid would have less need of a strongly adhesive foot in the shelter of a Frullania pitcher than in many more exposed situations. The discoid foot is an advantage to a species in any situation, and it is to be noted that all the species having it are large, powerful animals. Two species having the discoid foot are here described. One is from a lake, the other from a pond, and neither has yet been found on Hepatics. Formation of food-pellets. — Four of the species described in this paper belong to that section of the genus Callidina in which the food is moulded in the oesophagus into pellets. All the animals having this characteristic agree in many other points of structure, such as the small size of the corona, and form a very natural group. Most of them have the neck very long and the gullet correspondingly elongated, and forming a large loop when the neck is contracted. The pellets differ greatly in consistence in different species. Some are loosely put together, and quickly disintegrate when passed into the stomach. In some species they seem to be mixed, while in the oesophagus, with something which gives them coherence. Such pellets maintain their size and form unchanged during the whole of their passage through the alimentary canal, and are finally voided entire. When first passed into the stomach they are granular, and often dark from the admixture of food particles. They gradually lose the granular character as they move through the stomach, till when passed out they are clear spheres. 370 MR JAMES MURRAY ON A NEW FAMILY BDELLOIDA. All the Bdelloid Rotifera hitherto known have been included in two families, distinguished by different types of corona. The Philodinadse have the corona divided into two discs, which bear the primary and secondary wreaths of cilia. The Adinetadse have no discs, the corona consisting of a flat surface, furred with short cilia, divided by a non-ciliated space in the middle line, which may correspond to the space separating the discs in the Philodinadse. An animal discovered in Loch Vennachar in 1902, in the course of the work of the Lake Survey there, could not be referred to either of the known families. After prolonged study, continued for more than two years, it is now proposed to constitute a new family for its reception. Microdinad^e, n. fam. No corona, the ciliated alimentary tract ceasing at the mouth ; jaws intermediate between the ramate type of all other Bdelloida and the malleo-ramate type of Melicerta. The only species at present known is a Philodinoid animal. It resembles the genus Philodina in general form, in the rostrum, and in having four toes. The absence of corona would not of itself have justified the establishment of a new family. It might have been regarded as a degenerate Philodina which had taken to a different mode of feeding, and lost its corona from disuse. It was only after the peculiar structure of the jaws was understood that it became evident that the definition of the family Philadinadse could not be modified to include it. As now understood, the new family is seen to differ more from the other two families, of the order than they do from one another. The Adinetadse differ from the Philodinadse only in the form of the corona, and in the partly adnate rostrum, free at the tip. They have the same form of jaws and of all other structures. The Microdinadse differ from both, not only in the lack of corona, but in the shape of the jaws. It comes nearer to the Philodinadse in the free rostrum and the number of toes. On the other hand, the form of mouth might more readily be derived from that of Adineta. In Plate II. are shown heads of Philodinadse (fig. a), Adinetadse (fig. c), and Microdinadse (fig. 6). They are drawn from the ventral side in order to show the similar form of lower lip in all. On the same plate are drawn three pairs of jaws : — fig. d shows the ramate jaws of Philodinadse and Adinetadse, fig. e those of Micro- dina, fig. f those of Melicerta. It will be seen that the jaws of Microdina differ about as much from the ramate as from the malleo-ramate type, and sufficiently approach the latter, in the anterior position of the teeth and the less rigid union of the various parts, to constitute in some degree a link between the Bdelloida and the Rhizota. AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 371 Microdina, n. gen. Toes, four. Yolk-mass with four nuclei. Gullet very short. Teeth, three or four on each side, at anterior end of jaws. The terminal cilia of the gullet, which project a little way and assist in seizing the food, might be regarded as constituting a rudimentary corona. It is not usual, how- ever, to regard the end of the gullet in a Philodine as part of the corona, that term being restricted to the discs and the two wreaths. It seems, therefore, more correct to consider Microdina as having no corona. The parts of the jaws are movably articulated, not rigidly united as in other Bdelloids. The rami have large curved processes on the ventral side. The manubrii may have no loops, or may have from one to three, more or less distinct. In Philodinadse and Adinetadse the larger teeth cross about the middle of the jaw, and there are finer strise towards each end. In Microdinadse the large teeth are all at the anterior end, and fine strise only cross the posterior half of the jaws. Owing to the shortness of the gullet, the jaws are close to the mouth. Microdina paradoxa, n. sp. (Plates I. and II.) Specific characters. — Of moderate size, stout, enlarged at level of mastax and at posterior part of trunk, contracted between those parts. (Esophagus and large granular mass connected with it, of a bright crimson colour. Stomach voluminous, its walls filled with coloured globules. Foot of three joints. Spurs short, stout, somewhat bottle-shaped, a broader basal portion contracting into a narrower apical portion, obtuse, separated by narrow convex interspace. Toes large and thick, the ventral pair much larger than the dorsal. Foot-glands forked. Antenna two-jointed, flattened. Oviparous. General description. — Greatest length y^ to g^ inch. Always fiddle-shaped, owing to the narrowing between the head and the enlargement of the trunk. The position of the posterior enlargement varies. When an egg is carried, the broadest part will be in the second or third central segment. When there is no egg and the ovaries are undeveloped, the fourth central or pre-ancd may be broadest. The rostrum differs in no way from that of a typical Philodine. The mouth is small and somewhat trifoliate. The lower lip is of the V-shape usual in the order. The upper part is obscurely two- lobed. The sides of the mouth are finely longitudinally striate. The whole animal is sometimes pale rose-colour or purple. More generally it is colourless, except for the crimson gland and oesophagus, the stomach and the egg. The globules in the stomach walls vary greatly in colour. They have been seen red, yellow, greenish, orange, magenta, sienna, or umber. The two last colours are commonest, and are used in the illustrations. The egg is of a tawny yellow. A clear fleshy mass fills the head 372 MR JAMES MURRAY ON A NEW FAMILY between the rostrum and the oesophagus. The posterior portion of this mass is, from its position and its connection with the antenna, regarded as the brain, but its outline could not be traced. Between the mastax and the stomach are two clear gastric glands, which meet on the ventral side. Habits. — Of tireless activity. It creeps without ceasing on the stems of algae and mosses, feeding all the time. Its mode of feeding is unlike that of any other Bdelloid, though Adineta resembles it in some respects. A biting action is continually repeated. In this the rostrum takes part. The food is caught between the rostrum and the lower lip, and pushed close to the mouth by the bending down of the rostrum. The cilia of the mouth, working downwards, catch the food that is thus brought near and sweep it into the gullet. The brush of cilia on the rostrum contributes to the action by sweeping downwards also, and to some extent compensates for the lack of discs. It was never seen to pause or rest, as other Bdelloids do occasionally. The deposition of the egg was on one occasion seen. The animal was fully con- tracted. When the egg was almost completely passed out, the end which still remained in the aperture was seen to be surrounded by a circlet of clear spherical bodies. Most of these adhered to the egg when it separated, but a few remained attached to the aperture (Plate I. fig. d). Variation. — Only one species of the family is known with certainty. The lack of corona deprives us of several characters of great service in distinguishing species of Philodinadge. Various forms of spurs have been seen in Microdinadae, but it is not yet clear whether any of these belong to distinct species (Plate II. figs, g toj). The jaws also differ in different examples. While agreeing in general features, the degree of development of the loops of the manubrium varies greatly in different individuals. Some show no trace of any loops, while others have three well developed, two on the outer side and one on the inner, passing behind the teeth. Habitat. — At the margins of large lakes and of clear hill lochs, also occasionally in pure running water. Discovered in Loch Vennachar, 20th May 1902, on the occasion of the visit of the Scottish Natural History Society, as guests of Sir John Murray ; Loch Ness and Loch Morar, 1903; hill lochs on Carnahoulin, Fort-Augustus, 1904; Loch Treig, December 1904. Very abundant in Loch Vennachar and frequent in Loch Ness. Philodinadge. Classification. — The Ehrenbergian division of the Philodinadae into genera dis- tinguished by the presence or absence of eyes, and by the position of the eyes when present, has long been recognised as artificial. In those genera unrelated species are brought together, and closely related species are separated. Suggestions for a more natural classification have been made, notably by Milne, but none have been generally AND TWELVE NEW SPECIES OF ROTIFER A OF THE ORDER BDELLOIDA. 373 accepted. The number of known species belonging to this family is now so great that some subdivision of the genera would be desirable, even if those genera were natural. Many of the new species show the artificial character of the old genera, and render a revision imperative. I understand that a revision of the genera is now being prepared by Mr Bryce, who, from his long experience of the order, is so well qualified to do so. This being so, I shall here only amend the definitions of the genera Philodina and Callidina so as to render them more natural. The classification based upon the eyes having proved defective, other characters of a more reliable nature have been sought. The number of toes has been suggested by Milne * as a basis for classification. The mode of repro- duction was thought of. It was found that large groups of species agreeing in the number of toes, also agreed in the mode of reproduction. One or two exceptions, however, lessen the value of the mode of reproduction as a generic character, and it must be abandoned in the meantime. Philodina. Generic character. — Toes, four. Milne's suggestion is adopted, though it is recognised that the genus will have to be divided. Thus defined, the genus does not differ greatly from that of Ehrenberg. All the species having eyes in the neck {i.e. seated on the brain) are found, with one exception, to have four toes. The main result of the alteration will be the transfer to Philodina of several species hitherto included in Callidina. Callidina. Generic characters. — Toes, three ; or foot ending in a disc. Normally oviparous. This definition is simply provisional. It is unsatisfactory, in that it includes a character, viz., the mode of reproduction, which is not quite invariable. It is only by using this character that the genus Rotifer could be kept separate. As formerly distinguished by a single negative character, viz., the absence of eyes, the genus Callidina became the receptacle for all the homeless wanderers of the family, till it now includes a host of species, many of which have little affinity one with another. It is with this genus that a revision of the family will be mainly concerned. Four of the new species here described belong to that very natural section of the genus in which the food is moulded into pellets. Two have the discoid ' symbiotic ' foot. This type of foot might be made the basis of a genus, were it not that it is in some cases impossible to determine whether there are separate toes or not. It is, moreover, suspected that the discoid foot may have been independently acquired by unrelated animals. * Proc. Phil. Soc. Glasgow, vol. xvii. p. 134, 1886. 374 MR JAMES MURRAY ON A NEW FAMILY Callidina angusticollis, n. sp. (PL III. figs. 2a to 2k.) Specific characters. — Small, colourless ; form pitcher-shaped in lateral view, the lower lip large, elevated, spout-like. Discs small, close together, inclined obliquely towards the mouth. Oral segment elongate, encircled about midway by a series of four thickenings. Food moulded into pellets. Foot minute, not obviously segmented ; spurs short, acute, decurved, meeting at base. Dental formula 2/2. Secretes a brown flask- shaped protective case. General description. — Greatest length -^ to -j-^ inch when feeding. Head laterally compressed, elongate from front to back. Discs sloping downward and outward from middle line as well as forward towards the mouth. Lower lip larger, relatively to the size of the animal, than in any other species known. Thickenings on oral segment diagonally placed, as shown in section, fig. 2e. Rostrum of moderate length, with fairly large lamellae. Antenna of two joints, length equal to f diameter of neck. Neck with large rounded thickenings at each side of antenna, and ventral thickening. Neck very long and slender. Gullet correspondingly elongated. Stomach voluminous, filled with round, clearly -outlined pellets of uniform size. These are coherent, and do not disintegrate in their passage through the alimentary canal. They are voided whole. No eyes. Reproduction oviparous. Case oval, slightly flattened on ventral side, pale yellow when young, dark brown when old. Neck of case long, with annular striae, mouth slightly expanded. The foot, being apparently useless inside the case, which the animal never seems to leave, is very small. It can only be seen when the animal is forced out of its case. No separation of the first and second joints can be distinguished. The rudiments of toes probably exist, as the spur-bearing joint is not closed at its lower end, but they were not seen. Habits. — Trusting apparently to the protection afforded by its shell, it is not at all shy, and usually resumes feeding very soon after being disturbed. When feeding, the neck is bent backward. Before beginning to feed, the head is often put out and the neck bent sharply over the edge of the case till the rostrum touches the outside of the case. The case is believed to be secreted from the skin, but the process has not been observed. The animal may occupy empty (or even inhabited) shells of Rhizopods, such as Difflugia or Nebela. Careful examination has always revealed a normal case inside the shell thus occupied. On one occasion the Callidina was seen in a shell of Difflugia which was shorter than its case. The projecting neck of the case was viscous, as shown by adherent matter, and nearly colourless. It had probably been just completed. Old animals show no viscosity, either of skin or case. The case is thin, smooth, and brown, and does not adhere to the animal. It is a cleanly animal. The pellets, which at first contain the food, are eventually passed out as clear spheres. After voiding them it clears them out by fully contracting its body and rolling about from side to side of the case till they are forced out through the neck. AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 375 Habitat. — On the leaves of mosses and hepatics in a great variety of situations, in peat bogs, on the ground, walls, or trees, most frequently seen on Fontinalis growing at the margins of lakes. First seen in Loch Morar, common about Fort- Augustus ; occurs also in North Uist. Probably widely distributed. Before the animal came under my notice Mr Bryce had made some studies of it, and suggested the specific name. Dr Penard has also studied it in Switzerland. Callidina annulata, n. sp. (Plate III. figs. 3 a to 3/) Specific characters. — Small, colourless, in lateral view pitcher-shaped, the lower lip spout-like. Discs inclined towards mouth, their surfaces parallel to long axis of body. Oral segment much elongated, marked by annular plicae, which are stronger towards the base. First neck segment with similar plicae. Antenna very small, its length equal to f of the diameter of the neck. Teeth, seven or eight in each jaw. Food moulded into pellets. Foot short, of three joints. Spurs, short cones, meeting at base. Repro- duction oviparous. General description. — Length about T ^ inch when feeding. Oral segment twice as long as broad. Discs reniform, separated by very narrow sulcus. Neck and gullet very long. Rostrum short and broad, with small lamellae. Stomach large, nearly filling the trunk, containing clear rounded pellets of uniform size. Resembling C. angusticollis in size and general form, it may be easily distinguished from that species by the smaller lower lip, greater forward inclination of the discs, longer oral segment, with annular plicae and without thickenings, numerous teeth, larger foot, and lack of protecting case. Some examples carried large oval eggs. Intestine, glands, and vibratile tags were not observed. Habits. — Being unable to secrete a case for itself, as is done by C. eremita and other species, it seeks shelter, like the hermit crab, in the empty shells of other animals. Shells of Diffiugia, Nebela, and other Rhizopods are commonly occupied. It was first observed in cases of C. angusticollis, the original occupants of which had died, leaving only the tough jaws behind. The presence of those jaws, with their pairs of teeth, led to the two species being confused for some time. It also frequently takes cover in the pitchers of Frullania and other Hepatics. It is often found creeping about without protection of any sort, but it has never been seen to feed unless when in a shelter of some kind. When feeding it is not timid. It may frequently be observed, in detached pitchers of Frullania, whirling rapidly about, regardless of collisions. Habitat. — Among aquatic mosses growing in Loch Morar, October 1903, Loch Ness, 1904. Not confined to lakes. Common on Hepatics, Fort- Augustus, Blantyre Moor. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 15). 55 376 MR JAMES MURRAY ON A NEW FAMILY Callidina crenata, n. sp. (Plate IV. figs. 6a to 6d.) Specific characters. — Small, colourless. Trunk and foot papillose. Head and neck smooth. Neck with a prominence on each side of the antenna. Teeth, seven or eight in each jaw. Foot of three joints. A crenate boss on first joint. Spurs short, tapering, acuminate, divergent. Toes, three, Food moulded into pellets. Posterior margin of pre-anal segment with a rounded prominence, free from papillae, on each side. Oviparous. Description. — Length gV i ncn when fully extended. Rostrum short, with lamellae of moderate size. About twelve longitudinal folds on the trunk, at equal distances apart, not fainter dorsally. Papillae not crowning the folds, as in C. aspera, Bryce, but regularly distributed over the whole surface of the trunk, smaller than in C. aspera, rounded, without pits or pores, diminishing in size on the foot. Spurs dotted. Toes short, blunt. Egg elongate, narrowed at anterior end. This description is incomplete, as the animal was never seen to feed. Seen in the retracted state the discs are small and close together. The only other species which moulds the food into pellets, and at the same time has the skin papillose, is C. aspera, Bryce. From that it is distinguished by the more numerous teeth, smaller papillae, and pre-anal processes. Habits. — Although fairly abundant in several collections, nothing could be learned as to its habits. All the examples studied were very sluggish in their motions. They crept about very slowly ; and though some of them were watched for long periods, they showed no disposition to feed. Habitat. — Among ground moss and hepatics from the shores of Loch Ness and elsewhere near Fort- Augustus, February 1904, frequent; not yet found anywhere else. Callidina pulchra, n. sp. (Plate IV. figs. 5a to 5f. ) Specific characters. — Small, colourless. Trunk very broad, strongly stippled. Corona narrower than neck or collar, with central setae on discs. First neck segment with the anterior edge turned outwards like a rim all round. Rostrum short and broad, with a large brush of long cilia. Teeth, three to five in each jaw. Food moulded into pellets. Foot short, of three joints. Spurs short, divergent, acuminate. Toes, three. General description. — Length about y^ inch when creeping, ^-q inch when feeding. Very short and broad. Skin not papillose, but covered with uniform large clear dots. Trunk longitudinally plicate ; dorsal folds faint, lateral deep. Stomach very voluminous, filled with large pellets. Very similar to C. lata, Bryce, to which it is closely related. It agrees with that species in the breadth of trunk, the shape of the corona, the central setae on the discs, and the dental formula. It differs in the oval rather than ovate trunk, the stippled skin, the projecting edge of the first neck segment, and the shorter spurs. The shape of the AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 377 trunk makes it a more elegant animal than C. lata, the great posterior breadth of which imparts a clumsiness of gait as it moves. Habits. — In its steady, deliberate motions and mode of feeding it resembles C. lata. When creeping it goes steadfastly forward, increasing the length of each step by a glid- ing movement produced by the cilia of the rostrum. It feeds for shorter periods than C. lata. Habitat. — In ponds near Fort- Augustus, February 1904. Found among the sedi- ment obtained by washing aquatic mosses. It was very abundant in some ponds. When a portion of the sediment was put into a bottle with some water and tightly corked the animal continued to abound, and increased in numbers for some months, though the water was never changed. Callidina muricata, n. sp. (Plate V. figs. 7a to 7h.) Specific characters. — Of moderate size, narrow. Trunk with strong longitudinal plicae, covered with low rounded tubercles. Corona narrower than neck. Discs large, separated by very small interstice. Rostrum narrow, with large lamellae, which project laterally. Antenna slender, length equal to half diameter of neck. Neck with large thickenings on each side of antenna. Brain large, elongate ; no eyes. Dental formula 2/2 ; border of jaws crenate. Food not moulded into pellets. Foot short, of four joints. Spurs slender, tapering, meeting at base, divergent, incurved. Toes, three ; large, taper- ing. Reproduction oviparous. General description. — Greatest length -^ to -j-^q inch. Skin of trunk yellowish, viscous, with little extraneous matter adhering. Stomach large, its walls containing small dark-greyish globules. Yolk-mass with eight nuclei. Egg large, oval. Discs nearly touching. Border of jaws brown. The tubercles are of equal size, rounded, and disposed in transverse and longitudinal rows. They are probably permanent, and not mere hardened secretions as in C. incrassata, but this is not proven. On the back they are hidden by the deep longitudinal plicae. The transverse rows, about nine on the trunk, give a false appearance of close segmentation. The tubercles are more obvious on the ventral side, and all over when fully retracted. The glands, intestine, and cloaca were normal. Vibratile tags not seen. Apart from the tubercles, the species may be known by the close approximation of the large discs and by the caliper-like spurs. Habits. — Very slow in its motions. It extends itself with studied deliberation, like Rotifer tardus, and is not often willing to feed. It feeds steadily, but only for a short time. On all the occasions when it was seen feeding the ventral side was uppermost, so that the details of the upper lip could not be seen. Habitat. — In the sediment of ponds, Fort- Augustus, January 1904, frequent; Blantyre Moor. 378 MR JAMES MURRAY ON A NEW FAMILY Callidina crucicornis, n. sp. (Plate V. figs. 8a to Sg.) Specific characters. — Large, slender, elongate. Rostrum very long, of two con- spicuous joints, with very large, spreading lamellae. Antenna very small. Brain large ; no eyes. Jaws relatively very small ; dental formula, 2/2. Stomach voluminous ; food not moulded into pellets. Foot short, of three joints, very prominent dorsal boss on first joint. Spurs long, tapering, with distinct shoulder on inner side at base, capable of being brought together at the points or crossed over one another. Last joint of foot long, with three very large toes. General description. — Greatest length -g^ to -^ inch. Every part elongate except the foot. Colour dull yellow or greyish. Longitudinal plicae few, fainter on back. Salivary glands well developed, one long narrow pair extending beyond the mastax to the upper part of the stomach. Walls of stomach thick, filled with larger and smaller dark yellow globules. Intestine oval, its long axis transverse, partly covered in dorsal view by stomach. Yolk-mass large, with eight small nuclei. Space between spurs straight or convex, according to position of spurs. Terminal toes long, slender, two- jointed. Dorsal toe as long as the others, but usually extended to only half the length. Foot-boss pointing backwards. Owing to its disinclination to feed, the description cannot be completed. In the retracted state the discs are large and elongate. The species has a superficial resemblance to Callidina longirostris in the long rostrum and spurs, and also to Philodina macrostyla and its allies. It is believed to have no close affinity with any of those species, all of which are viviparous, while this is oviparous. The rostrum tapers gradually from the oral segment, and is not abruptly narrowed as in C. longirostris. Habits. — Although it has been known for more than two years, and has been under constant observation for nearly one year, and thousands of examples have been carefully studied by three or four observers, little is known of its habits, as it has never once been seen to feed. It creeps slowly and deliberately, examining everything it encounters with its rostrum, which appears to be a very delicate organ of touch. It is very mobile, and can be bent backwards and forwards and from side to side. The lamellae, which are only inferior in size to those of C. cornigera, are waved about in the way characteristic of that species, and which has led to the supposition that they are organs of smell. When washed out of the mosses among which it lives, and allowed time to settle down among the sediment, it is found that it takes up its position, not on the surface of the sediment, but a little way down in it. The stomach is often seen to be well filled with food. These facts, together with its disinclination to feed, lead me to suppose that it may have an aversion to light, and will not feed unless in darkness If this is so, it may be impossible to complete the description of the head. Against the AND TWELVE NEW SPECIES OF EOTIFERA OF THE ORDER BDELLOIDA. 379 suggestion is the fact that though it will not feed, it does not appear to be uneasy in the glare of the microscope lamp. Habitat.— In lakes and ponds. Bottom of Loch Rannoch, at depth of 9 or 10 feet, April 1902. Abundant in pond near Fort- Augustus, January 1904. It thrives well in tightly corked bottles, and may continue in them for months. Callidina armata, n. sp. (Plate VI. figs. 10a to 10/i.) Specific characters. — Large, massive. Corona broader than trunk. Rostrum short, broad ; lamellge small. Antenna as long as diameter of neck, clavate. A pair of tooth- like processes close below the mouth. Jaws relatively small, with two teeth on each. Stomach voluminous, reddish. Foot short, of three joints. A broad rounded fold at base of first joint. Spurs small, acuminate, incurved and decurved, interspace equal to diameter of base of spur. Foot ends in round perforated disc. General description. — Greatest length when creeping, -^ inch. Trunk with few longitudinal folds, dorsal faint, lateral deeper. All colourless, except alimentary canal. Food not moulded into pellets. Walls of stomach containing small reddish globules. Intestine roundish. Foot-glands of few cells, terminal cell largest. First foot-joint marked with annular striae. Terminal joint long, disc perforated by many pores, ducts in common sheath. Four pairs of vibratile tags were seen. Resembling C. symbiotica and allied species in massive build and discoid foot, it may be distinguished from all other species by the ventral processes below the mouth, the heavy antenna, and the dorsal fold at the base of the foot. Habits. — Strong and active, like all the ' symbiotic ' species. As it creeps rapidly about, the disc is exposed for an instant. It is a steady feeder. The function of the processes below the mouth could not be gathered from its actions. Habitat. — On water weeds growing in Loch Ness and the Caledonian Canal at Fort- Augustus ; although abundant during November and December 1903, it was not again found till December 1904, when it once more became common. The same beds of weeds, chiefly Myriophyllum and Fontinalis, were frequently examined during the intervening months without the species being once found. This may indicate that it has only a short season, though it is unusual for Bdelloids to have any seasonal limits. Callidina incrassata, n. sp. (Plate VI. figs. 9a to 9f.) Specific characters. — Large, stout. Trunk protected by thick plates formed of a hardened secretion. Rostrum short and very broad, with small ciliate lamellae. Antenna considerably longer than diameter of neck. Neck with large process at each side of antenna. Corona as wide as trunk, discs large, interstice equal to half diameter of disc. Central papillae on discs. Foot very short, of three joints. Spurs small, twice 380 MR JAMES MURRAY ON A NEW FAMILY as long as broad, acuminate, divergent, obtuse, incurved. Jaws with broad, brown, pectinate border, dental formula 5/4. Food not moulded into pellets. Oviparous. General description. — Greatest length -£q to -^ inch. Trunk dark yellow. Anterior row of tubercles more prominent than the others, sometimes so long that they hang down over the next two rows. Third segment of neck, close to tubercled trunk, viscous, and with a little extraneous matter adhering. Rostrum slightly broader towards apex, ciliated cup usually quite everted, the lamellae then standing far apart. Papillae on discs, only once seen, like little curved thorns. Viscera difficult to see through the dark, thickened skin. Under strong pressure stomach seen, with its walls filled with small clear globules. Brain large. Glands, intestine, cloaca, and vibratile tags not seen. Arrangement of teeth unusual. Three large teeth in one jaw fit into the spaces between four large teeth in the other. There is an additional thinner tooth at each end of the row of three. In the contracted state the tubercled trunk is so similar to that of Philodina mac- rostyla, variety tuberculata, that it might be passed over for that species. When it extends itself it is found to differ in everything else. Every part of the Philodine is long and slender, of the Callidine short and broad, except the antenna. This is straight, not elbowed as in P. macrostyla. The foot appears to end in a disc, as in the ' symbiotic ' Colliding. It is a very small and obscure disc, and no perforations could be seen. It may yet be found to have short, broad toes. The tubercles could be removed by rolling the animal under the coverslip. Habits. — Very slow and cautious. After being disturbed it may remain fully contracted and motionless for a long time. It puts out its head very gradually, feel- ing carefully about with its long antenna before venturing out. When it has gained confidence it walks forward rather briskly for an animal so heavily armoured. The very short foot is only momentarily seen, the disc not at all, unless it happens to be walking upside down. It was not eager to feed, and when it tried to do so was evidently annoyed by the debris surrounding it, and soon desisted. Habitat. — In the sediment of one or two ponds at Fort- Augustus, February 1904. It was pretty abundant for a time in one pond. Philodina laticornis, n. sp. (Plate VII. figs. 12a to 12c.) Specific characters. — Very large. Foot and rump together about f of greatest length when creeping. Corona narrower than trunk, discs with small central papilla?. Rostrum short, broad, with very small ciliate lamellae. Antenna stout, length equal to -f diameter of neck. Brain fairly large, with pair of large, oblique, yellowish-red eyes. Two teeth in each jaw. Foot of three joints. Spurs large, broad, divergent, interstice slightly exceeding diameter of spur at base. Dorsal toes small, ventral long, AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 38 1 incurved. Reproduction viviparous. Swims free, with spurs brought close together and discs also approximated. General description. — Greatest length -^ to ^ inch when creeping. All colourless except the alimentary canal. Longitudinal folds of trunk few, dorsal faint, lateral stronger. Brain elongate, triangular. Stomach ample. Intestine elongate, elliptical. Rump and foot of about equal length. First foot-joint long, with faint annular striae. Second joint with stronger striae, crossed above the spurs by two oblique folds of skin, which nearly meet in the middle line, and give the appearance of an extra joint. Spurs with obscure shoulder on inner side at base, then slightly contracted and ex- panded again nearer tip. Foot-glands rather small, with very long ducts. Most examples with two well-grown young, showing teeth and corona, and one younger foetus. Vibratile tags, five on each side seen. Habits. — The large size, lanky form, and large spurs and toes, suggest that the animal is a parasite, but it has not yet been found attached to any host. On the other hand, its readiness to swim, and its characteristic attitude when swimming, spurs and discs being brought together as though to lessen the resistance, are like the actions of a free-living animal. Several species of parasitic Bdelloids have small brains and are blind, and there is some reason to believe that this reduction is a consequence of the mode of life. The power of swimming might be of advantage to an ectoparasite by enabling it to change its host if necessary. When swimming, the rostrum is kept fully extended. When creeping, the toes are often kept out during the whole of the step. Habitat. — Among aquatic mosses growing at the margin of Loch Ness, at Fort- Augustus, April 1904 ; in the Caledonian Canal, Fort- Augustus, December 1904. Philodina laticeps, n. sp. (Plate VII. figs. 11a to 11/i.) Specific characters. — Very large, elongate, yellowish. Corona very large, much wider than trunk, discs broad, concave, separated by space nearly equal to diameter of disc. On each disc an elevated conical papilla, with broad apex bearing several short motile cilia. Rostrum short and broad, with minute lamellae. Antenna short, length equal to ^ diameter of neck. Brain a minute triangle, no eyes. Teeth, two on each jaw, with one thinner tooth. Foot and rump together just under half of total length. Foot of four joints. Spurs large, broad, blade-shaped, divergent, interstice equal to diameter of spur. Dorsal toes small, ventral long, incurved. Parasitic on insect larvae. Oviparous. General description. — Greatest length -^ to ^ inch when creeping. All hyaline except alimentary canal. Trunk longitudinally plicate, central segments covered with a hair-like growth, which is probably a vegetable parasite. Corona broadest and discs largest known in the order. Yolk-mass with eight or nine nuclei ; the large egg pointed at anterior end. Intestine long, elliptical. Foot-glands long, with very long ducts. Four vibratile tags on each side seen. 382 MR JAMES MURRAY ON A NEW FAMILY Habits. — Parasitic on insect larvae which live in running water. It has been found on larvse of several species, adhering to the thorax, between the bases of the legs. When separated from its host it is little disturbed, immediately begins to creep actively about, and readily feeds. When feeding it is very restless, and sweeps the great corona from side to side and all over the field. The apparent breadth of the corona is often in- creased by a peculiar habit the animal has of pulling in the sides of the trunk till it resembles a stem supporting a large flower. It is then more like one of the large- headed Ehizota, such as (Ecistes velatus, than a typical Philodine (fig. llh). When feeding it draws the rostrum in till it is depressed below the surrounding surface of the head. Comparison o/"P. laticeps with P. laticornis. — The two species resemble one another very closely in some characters, and differ greatly in others. The agreement is so close that it is difficult to avoid the conclusion that they are related animals. On this supposition an interesting comparison of the differences of structure in relation to the different modes of life may be made. P. laticeps is a parasite ; P. laticornis has only been found free. They agree in general form, in the rostrum, spurs, and long curved ventral toes, so closely that but for the longer foot of P. laticeps the same drawing of the extended animal could represent both. P. laticornis has a large brain and eyes, small papillae on discs, larger antenna, and shorter three-jointed foot. P. laticeps has much larger corona, very large papillae on the discs, shorter antenna, longer four-jointed foot, much smaller brain, and no eyes. If the parasite P. laticeps has been derived from the free-living P. laticornis, it is interesting to note that while it has gained a larger mouth, it has lost its eyes and most of its brain. Should P. laticornis, as is possible, prove to be also parasitic, the force of the comparison is diminished, but not altogether lost. The habit of swimming might enable a parasite to change its host when necessary, and so render it less dependent, and the retention of the large brain and eyes may be attributed to this habit. P. laticeps is oviparous, P. laticornis viviparous. This is the only instance known to me of closely related Bdelloids differing in the mode of reproduction. Habitat. — In a little stream entering Inchnacardoch Bay, Loch Ness. Very abundant during the winter of 1903-4. Any handful of Fontinalis taken from this stream and shaken in water yielded thousands of examples. Early in the summer of 1904 the stream dried up, and remained in this condition till October. When the water returned to the channel insects and rotifers had disappeared, and up till the end of November neither had again been found. Similar streams in the same district were searched, but though larvae were found, there were no rotifers upon them. Philodina humerosa, n. sp. (Plate IV. figs. 4a to 4 a.) Specific characters. — Small, dull grey, strongly plicate on trunk. Ventral trans- verse folds, fourteen or fifteen. Central setae on discs spring from large conical AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 383 prominences. Space between discs equal to half diameter of disc. Rostrum short, shaped like an acorn, basal joint papillose. At back of oral segment, on each side of rostrum, a large rounded papillose prominence. Length of antenna equal to diameter of neck. Neck with rounded prominence at each side of antenna. Foot of three joints, stippled. Spurs small, tapering, divergent. Dorsal toes tapering; ventral larger, obtuse. Two teeth on each jaw. Oviparous. General description. — Greatest length when creeping, y^- to t ^-q inch. Skin of trunk dull yellowish-gray, opaque, finely stippled, foot more strongly stippled. The prominence from which the central seta arises occupies nearly the whole of the upper surface of the disc. The great papillose bosses on the back of the oral segment are unique. They are conspicuous when the animal is creeping as well as when feeding. The skin of the first foot-joint is thickened dorsally, but does not form a boss. Related species. — This species is closely related to Philodina alpium (Callidina alpium, Ehr.) and P. brycei, (C. brycei, Weber). The three species form a very natural group. They are semi-loricated. The skin of the trunk is thick. Its anterior edge is •cut into definite forms and bears six knobs or processes. Its ventral surface is crossed by deep transverse folds, 9 to 15 in number. Though not quite rigid, it alters little in shape. When the animal is fully retracted the deep longitudinal folds allow the anterior edge of the trunk to be closed. In P. alpium and P. brycei the two anterior dorsal processes of the trunk form a fork which receives the antenna, as in Anurwa and Brachionus. In all three species the central setae rise from large conical processes. There are four toes. Habits. — Like its relatives P. alpium and P. brycei, it is very slow in its move- ments. When it has been left undisturbed for a time it feeds with confidence. It ceases feeding at short intervals, but resumes again at the same spot. Habitat. — Found in ground moss and Fridlania growing on stones. Old pier at the Monastery, Fort- Augustus, 7th February 1904. At several spots near Fort- Augustus. Not yet seen anywhere else. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 15). 56 384 MR JAMES MURRAY ON A NEW FAMILY EXPLANATION OF PLATES. The drawings of the complete animals are all made to a uniform scale, with the exception of Microdina,. which is drawn larger. The separate details are drawn of any convenient size. In the descriptions the only measurement given is the greatest length of the animal. All other measurements obtained are put into the drawings. Whenever possible the width of the corona, collar, neck, trunk, rump, and foot are measured, also the length of head, neck, trunk, foot, spurs, and jaws. These sizes, expressed in figures, convey but a vague impression of the appearance of an animal unless accompanied by a drawing. They are therefore omitted from the text. The form of the upper lip is carefully drawn, but is not included in the descriptions, as no common names for its various parts have yet been agreed upon. Plate I. 1. Microdina paradoxa, n. sp. a, dorsal view, example from L. Vennachar, 1902. b, lateral view, another L. Vennachar example. c, ventral view, variety from L. Treig, 1904. d, deposition of egg. e, /, antenna in different degrees of extension. g, foot, showing toes and glands under pressure. h, toes, dorsal view. i, rostrum, ventral side. Plate II. a, head of Callidina papillosa, ventral side. b, head of Microdina paradoxa, „ „ c, head of Adineta barbata, ,, „ d, jaws of Philodina brycei. e, jaws of Microdina, form with three loops. /, jaws of Melicerfa. g, h, i, j, four varieties of spurs of Microdina. k, I, m, n, o, jaws of Microdina, five views of same pair. k, direct ventral. I, oblique ventral. m, direct dorsal. n, dorsal, under pressure, rami turned on side. o, lateral. Plate III. 2. Callidina angusticollis, n. sp. 2a, animal in case, feeding, dorsal. 26, side of head, feeding. 2c, jaws. 2d, head seen from above. 2e, section of oral segment at thickenings. 2/, section of neck. 2g, spurs. 2h, side of foot and rump. 2i, animal in case, in characteristic attitude. 2j, side of rostrum. 2k, front of rostrum. 3. Callidina annulata, n. sp. 3a, animal in Frullania cup, feeding, dorsal. 3b, side of head. 3c, front of rostrum. 3d, jaws. 3e, antenna. 3/, spurs. AND TWELVE NEW SPECIES OF ROTIFERA OF THE ORDER BDELLOIDA. 385 Plate IV. 4a, dorsal view, feeding. 46, ventral view, creeping. 4c, back of head. id, side of rostrum. 4. Philodina humerosa, n. sp. 4e, section of neck. 4/, toes. 4g, jaws. 5a, dorsal view, showing stippling. 56, dorsal view, feeding. 5c, antenna. 5. Callidina pulchra, n. sp. 5d, side of foot. 5e, spurs. 5/, jaws. 6. Callidina crenata, n. sp. 6a, dorsal view, creeping, showing papillae. 66, ventral view, creeping, showing internal structure. 6c, section of neck. §d, jaws. Plate V. 7a, dorsal view, creeping. lb, ventral view, feeding. 7c, side of rostrum. Id, section of neck. Callidina murieata, n. sp. 7e, side of antenna. If, jaws. Ig, side of foot. 7h, spurs and toes. 8a, dorsal view, creeping 86, side of rostrum. 8c, front of rostrum. 8^, jaws. 8. Callidina crucicomis, n. sp. 8e, spurs crossed. 8/, side of foot. 8g, spurs and toes. Plate VI. 9a, dorsal view, feeding. 96, front of rostrum. 9c, jaw. 9. Callidina incrassata, n. sp. 9d, side of antenna. 9e, spurs and disc. 9/, section of neck. 10a, dorsal view, feeding. 106, ventral view, creeping 10c, side of head. lOd, jaw. 10. Callidina armata, n. sp. lOe, section of head, showing tooth-like processes. 10/, dorsal view of foot, showing fold and glands. \0g, side of foot. 106, spurs and disc. 386 NEW FAMILY AND SPECIES OF ROTIFFRA OF THE ORDER BDELLOIDA. Plate VII. 11a, dorsal view, feeding. 116, front of antenna, lie, side of antenna. \ld, papilla on disc. 11. Philodina laticeps, n, sp. lie, jaw. 1 1/, spurs and toes. 11 a, back of rostrum. 11/i, characteristic attitude, with trunk narrowed. 12a, dorsal view, swimming. 126, dorsal view, creeping, showing viscera. 12. Pliilodina laticornis, n. sp. 12c, side of antenna. Murray: Trans. Roy Soc. Edm r Vol. XLI. A New Family and Twelve New Species of Bdelloida.— Plate I. M'F»cU»t 4. Ermine. L.tlEdm MlCRODINA PARADOXA n. sp. Trans. Roy Soc. Ediif, Vol.XLI. fuRRAY: A New Family and Twelve New Species of Bdelloida.— Plate II. Trans. Roy Soc. Ediif, VbLXLI. Murray: A New Family and Twelve New Species of Bdelloida— Plate III. MfFa-rlane &. Ersfcme.Litl. Edm' 2, CALLIDINA ANGUSTICOLLIS.n sp. 3, CALLIDINA ANNULATA.n. sp. Trans. Roy Soc. Edm r Vol.XLI. Murray: A New Famiiy and Twelve New Species of Bdelloida.— Plate IV 4. M'FaxlaTie &. Er shine. Uth.ZAm* HILODINA HUMEROSA.n.sp. 5,CaLLIDINA PULCHRA,r,sp. 6, CALLIDINA, CRENATA.u.sp. Trans. Roy Soc. E diif. Vol. XL! Murray: A New Family and Twelve New Species of Bdelloida.— Plate V. 7 CalLLDINA MURICATA.n.sp. 8, CaLLIDINA CRUCICORNIS, n.sp MrFa.rla.-ne & Erskine, EiLkEdit 1 Trans. Roy Soc. Edirf, VoLXLI. Murray.: A New Family and Twelve New Species of Bdelloida.— Plate VI. 9, CALLIDINA INCRASSATA.n. sp. 10, CALLIDINA ARM ATA n. K!F»lui( UrAine.IitTi.Edu sp. Trans. Roy. Soc. Edirf, Vol. XLI. Murray.: A New Family and Twelve New Species of Bdelloida.— Plate VII. 11, PHILODINA LATICEPS.n.sp. 12, PhILODINA LATICORNIS, n. KFaTlans WSrskme litt Edin 1 sp. ' ( 387 ) XVI. — The Eliminant of a Set of General Ternary Quadrics. — (Part III.) By Thomas Muir, LL.D. (MS. received December 12, 1904. Read! January 23, 1905. Issued separately April 15, 1905.) (41) The in variance of the equations a Y i: 2 + b^j 2 + c x & + f x yz + y x zx + \xy = , \ a 2 x 2 + b 2 y 2 + c 2 z 2 + f 2 yz + g. 2 zx + h. 2 xy = , V a B x 2 + b 3 y 2 + c 3 z 2 + f s yz + </ 3 zx + h 3 xy = , J with regard to the group of cyclical substitutions, and the consequent invariance of the eliminant with regard to the reduced group con- sisting of the last two substitutions, has been already referred to. When the eliminant is expressed in terms of the three-line determinants formable from the array of coefficients, the invariance in question is self-evident, as each of the twenty-eight parts composing the expression is invariant by itself. For convenience this form may be repeated from § 31 with a slightly improved notation. It is 0U00 + 2l'4'8'9- + 20077' + 24457' -220016 + 21'55S' + 207'8'9' -21688' -220147' - 244'99' + 0456 -21T88' + -'20159 -21'6'88 + 0123 + 21137 + 20125' + 21157' + 204'5'6' - 24468 + 0'7'8'9' -21'448 - 0'456 + 21166 + 24o'7'9' -21489 + 213'7'S' -2H4'9 TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 16). 57 388 DR THOMAS MUIR ON THE where are used for 1 2 3 4 5 6 7 8 9 1 «A C 3 1; 1 1 a A/ s \, 1 Vi^S li 1 | c 1 a 2 h 3 1 <hh9e li | ftjCg/tg | , 1 c i a 2/s 1 ! «i V'a 1 > 1 V2/3 1 ! 1 ^ft | 1' 2' 3' 4' 5' 6' 7' 8' 9' 1 UiA 1. \ c \9Jh l> 1 V's/s 1 ' 1 & l/ 2 .?3 1 . 1 c A/a 1 » 1 «i/2#a 1 > I^A li K/203 l» 1 a i9i^% 1 > IV^/s ll respectively. (42) There is, however, a second form of invariance which it is convenient now to consider. Looking at the equations we at once see that the performance of the interchange Tx a f\ \ y b g ) leaves them unaltered, and that the same is true of either of the interchanges 71* c \>x a y b g\ ~pz c h z c h From this it follows that the eliminant is invariant to each one of the three interchanges ia f\ pb g\ 7K b g J , \ c h J , ba f Taking the first of these and observing its effect on our twenty determinants of the third order we find that it is equivalent to the substitution 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; 0', 1', 2', 3', 4', 5', 6', 7', 8', 9' 0, -4,-6,-5, -1,-3,-2, -7,-9,-8; -0', - 4', - 6', - 5', - 1', -3', -2', - 7', - 9', - 8' In an expression, like the special form of eliminant given in the preceding §, where each term is the product of four of these determinants, and where, therefore, the sign of each determinant may be changed with impunity, this substitution has the same effect as the simpler substitution 1 , 2 , 3 , 4 , 5 , 6 , 8 , 9 , 1', 2', 3', 4', 5', 6', 8', 9' 4 , 6 , 5 , 1 , 3 , 2 , 9 , 8 , 4', 6', 5', 1', 3', 2', 9', 8' or the interchange Similarly we have 1 , 2 , 3 , 8 , 1', 2' 3', 8' 4 , 6 , 5 , 9 , 4', 5' 6', 9' Tbff\ ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; 0', 1', 2', 3', 4', 5', 6', 7', 8', ?| , - 6 , - 5 , - 4 , - 3 , - 2 , - 1 , - 9 , - 8 , - 7 ; - 0', - 6', - 5', - 4', - 3', - 2', - 1', - 9', - 8', - 7' c:o-c ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 389 and therefore in the case of products of four 1,2,3,4,5,6,7,9; 1', 2', 3', 4', 5', 6', 7', 9' 6, 5, 4, 3, 2, 1, 9, 7; 6', 5', 4', 3', 2', 1', 9', 7' a, 2,3, 7; 1', 2', 3', 7'\ -6,5,4,9; 6', 5', 4', 97 ' and, lastly, in the same circumstances c h\ \fl, 2,3,7; 1', 2', 3', 7' a f) " \5, 4, 6, 8; 5', 4', 6', 8' (43) A comparison of the three interchanges, which, in the case of a four-factor product, we have thus found to be equivalent to Ta A fh g\ te h\ \,b gj, \c h), \,a f) respectively, leads at once to the further observation that if the expression in which the interchanges have to be made be invariant to the cyclical substitution, the three interchanges are not essentially different. So far, therefore, as the above eliminant is concerned, we need only consider one of the interchanges, say the interchange /1,2,3, 8; 1', 2',3',8'N U , 6 , 5 , 9 ; 4', 6', 5', 9' ) , it being borne in mind that this implies that the determinants 0,7; 0', 7' are invariant to the interchange. The determinants 0, 0', which are invariant to the cyclical substitution as well, we shall therefore speak of as being doubly-invariant. (44) Turning then to the eliminant and applying this interchange to each of its twenty-eight parts, we find that twelve of them, viz., the 1st, 2nd, 3rd, 4th, 5th, 7th, 11th, loth, 18th, 21st, 26th, 27th are doubly-invariant ; that twelve others may be grouped as six binomials which are doubly-invariant, either term of each binomial being produced from the other term, viz. 6th and 8th, 13th and 14th, 16th and 22nd, 17th and 23rd, 19th and 20th, 25th and 28th ; and that the four remaining parts (the 9th, 10th, 12th, 24th) are 20125', 04'5'6', -0'456, -24468. Now we can show (see §39) that 04'5'6' — 0'456 is expressible as the difference of two terms which are each doubly-invariant, viz., the difference 07'8'9' - 0789. Further, since 20125' = -21129 + 21246 390 DR THOMAS MUIR ON THE the other two terms are expressible in the form -(21129 + 24468) + 21246, where the binomial and the single term which follows it are both doubly -invariant. There is thus finally obtained an expression for the eliminant which shows its property of double-invariance, the constituent parts being fourteen single terms and seven binomials, viz. 0000 + 0123 + 0456 -2-20016 -21129 - 24468 + 20077' + 21137' + 24457' -2-20147' -2H4'9 -2T448' + 2-20159 + 2U5'7' + 21'558' + 4- 07'8'9' -211'88 - 244'99' 0789 + 2137'8' + 2457'9' + 0'7'8'9' + 21166 - 21489 + 2l'4'8'9' - 21688' - 21'6'88' (45) Let us return now to the eliminant of the 10th order as obtained in § 9, viz. «i A ffi h i A 9i A «i h \ ih • 9i h A a 2 A C 2 h. '.1-2 \ 0i A CLc, • h. 2 h G S 9 S ^3 A «3 A C 3 • h s 9 3 h fh A «3 h 6 4' 5' 6' + 0' By transposition of rows and of columns, and by altering its sign, this is readily changeable into 6' 1 c l *i 9i • A 2 '*2 h 92 A 3 C 3 *■ 9z A 1 °i h c i A 9i 2 a 2 h C 2 A r/ 2 3 «3 h C 3 A 9z 9j A a i h c i h 9 2 A a 2 h C 2 K ■ 9 3 A «3 \ C 3 h. ' 6 4 . 4' 5' 5 + 0' ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 391 — a form more suitable for obtaining the cofactors of the elements of the last row in terms of the familiar three-line determinants 0, I, 2, ... . Taking first the cofactor of + 0', and using Laplace's expansion-theorem, we find it = 0(00-16)-7(-89') + 4(02 -17') -3(05 + 69') + 9(78) - 8'(25 - 7'9'), = 000- 2016 -2147' + 2-789 + 7'8'9'. Similarly the cofactor of 4 is found to be -002 + 038 + 056 + 03'6 - 189 -l'48 + 224 - 255' + 3'47' - 4'99' - 559 + 5'7'9' + 669' + 6'89, and the cofactor of 4' to be -004+ 013 +036'+ 067 -119 -11'4+1'8'9' + 244 + 26'8' - 25'7 - 2'99' + 339' + 3'79 - 579 . The full eliminant is thus (0 + 0')[000 - 201 6 - 2147' + 2-789 + 7'8'9'] 002 + 038 + 03'6 + 056 - 1'48 - 189 + 224 255' + 3'47'-4'99'-559- + 2*' -004+ 013 +036'+ 067 - 119 - 11'4 + 1'8'9 - 1'48 -189 + 224-| ' + 5'7'9' + 669' + 6'89j + 244 + 26'8' - 257 - 2'99' + 339' + 3'79 - 579 This does not, of course, differ from the form used in § 41. As a matter of fact, it will be found on examination that nineteen of its thirty-eight terms agree with terms in the expression of § 41, and that the other nineteen can be changed without much difficulty so as to establish the identity of the two expressions. (46) As may be supposed, however, the importance of the new result does not consist in its affording a verification of that previously obtained. It is more interesting, in fact, in its unsimplified state ; for it has now to be noted that each of the three lengthy expressions found in it as the cofactors of + 0', 4, 4' can be put in the form of a simple three-line determinant. For example, 58' + 67 5 + 69' 48 6 59 4 9 A 47 ' 0+ ir + 58 o« + *1 . « . 5 1 + 789 5 6 4 ±«)t. O^' + f + £) + 0(^ + 5 -"' + «T) ; , ,,, ,59 Z A »u» _ V* ^699' = 000 + ^05 08' - 79 i>f 08' - 79 4 J ' ^"\ 4 = 000-2053-2369' + 7'8'9' + 2-789 , = 000-2016-2147' + 7'8'9' + 2-789, = cofactor of 4- 0'. + 7'8'9' + 2789, 392 DR THOMAS MUIR ON THE Similarly it may be shown that + 8 59 4 2- 55 ' 4 ►? 9 o 66' 3 " 5 48 '6 + 47 ' (i 6 and -= - 002 + 038 + 03'6 + 056 - 1'48 - 189 + 224 - 255' + 3'47' - 4'99' - 559 + 5'7'9' + 669' + 6'89 = cofactor of 4 ; + 28' 27 22' 5 "T h 3 3 9 o + ™ 6-f 19 2 7 *-T = -004 + 013+ . . . . = cofactor of 4'. It consequently follows that the eliminant may be put in the form (0 + 0') + 58' 8 59 4 O -2* 8 59 4 55' -2*' < 27 3 •-7 67 5 < 9 + f 5 9 3--? 9 n 39 ' 0+ 1 <t 33 ' 6- f 7 48 6 < 48 "6 47' + lT -T 19 2 7 *"T or, say, for convenience of future reference, (O + O^Aj - ±4A 2 - ±4'A 3 . (47) A glance at the cofactor of + 0' as first obtained, viz., 000 — ^016 — . . . . shows that it is invariant to the interchange of § 42. Since, therefore, the performance of this interchange on the determinant form of it, A 1? — where the in variance is not in evidence, — cannot make any alteration in substance, we shall obtain thereby a second determinant form V l5 thus arriving at the unexpected identity + 58' 59 "4 67 5 + 48 6 69' + 47' 04 9 ' 9 38 1 = 27 3 A 28 ' 8 7 19 2 o + ; Further, since the whole eliminant is invariant to the interchange, the removal from it of the invariant portion (0 + 0') A T must leave a portion which in substance, if not in form, is likewise invariant. This consideration gives us the identity ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 393 or, say, 2 4 2i 8 59 4 55' 2 ~T + 2 4 ' •Tt 27 3 3 + 6 / 5 9 o 66' 6 ~~5 9 + «T 6-f 48 6 < >4 4 ' 19 2 7 *-¥ 9 38 1 . 33' 6- T o + 2»' 5 67 5 3-f n 28 ' + T 8 22' 5 ~T 8 »4 8 ' 2- 55 ' 4 19 n *?' „ n ' 48 i 44 ' 2 + ¥ 4 "T 6 " 7 X ~T ±4A., + ±4'A, = £lV„ + ±1'V„. If we denote the first portion of the eliminant, viz. (0 + 0') A : by A, and the second portion by - B, and the alternative forms of these by A' and - B', we can thus express the eliminant in four different ways, viz. A-B, A-B, A-B', A-B'. (48) Using the cyclical substitution on the right-hand member of the immediately preceding identity, we see that we can put in place of it 2> 7 19 2 *"T + 22' < 48 "6" -f 39' -i — ' 1 9 . 33' 6 ~ 1 9 A 69 ' 0+ 5 o 66 6 ~ 5 27 < 22' 59 8 2-™ 4 3 5 -~3 4 But the determinants here are those occurring in the left-hand member : consequently we deduce ±(4-2'A 2 = ±(2-4')A 3 . (49) Returning to § 46, and noting that A 2 has two columns in common with A ]s and that the result of the cyclical substitution on A 2 is simply to change this pair of columns into another pair of A l5 the third column remaining all the while unaltered, we see that (0 + 0')A 1 - 24A 2 394 DR THOMAS MUIR ON THI can be put in the form a 58' + T 8 59 4 2-*- 5 ' 4 67 5 0+£ 5 9 „ 66' 1~1 7 48 6 < i-i 4 ' 6 4 5 6 + 0' A new expression for the eliminant is thus < 7 48' 6 -2*' a 2 8' + T 27 3 B .y 59 4 4 8 4 9 o + * 6 "T 9 67' 5 5 66' d 5 19 2 7 4-L 1 ' 2 6 4 5 + 0' This can be further improved by removing from the four-line determinant the terms containing 0', and associating them in the form 0' V x with — 24'A 3 just as 0\ has been associated with — 24A 2> the final result being + 47' 7 48 6 -¥' °4 r 19 2 59 4 < 8 2 -A 5 ' 4 + 8 -f 9 67 5 < o 66' 3 "¥ 38 1 9 6 4 5 4' 5' + 27 3 39' 1 6' 4- 6- 11' 2 22' 3 33' 1 0' From this, by making the now familiar interchange, we obtain the alternative form + 17' 7 19 *-¥ •4 r 48 7 i 44 ' h J 2 "6" ] -fi 38 1 A 39 ' o+ r 9 .-»' + 9 69' j — 5 67 5 o 66' d ~ 5" 8 27 3 28' + 3 -f 59 4 8 4 2-55' 4 2 1 3 1' 3' 2' 0' Here, however, if in both determinants we transpose the second and third rows, and thereafter the second and third columns, we find that either determinant has three rows in common with one of the determinants of the form from which this was derived. Subtraction is thus readily accomplished, the result being ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 395 < 7 48 6 '4 59 4 °4 8 ' 8 ■) 55 ' " 4 9 67 5 °4 „ 66' 5 6-1' 4-2' 5-3' + 17' 19 7 *4 *-T 2 8 A 28< 0+ y 27 3 22' 5 -y 38 1 9 . + * a 33' 6- y -.4' 3-5' 1-6' This is a verification of the identity ±(4-2')A 2 = ±(2-4')A 3 already obtained in § 48. The latter form of it shows that both of the four-line deter- minants are invariant to the cyclical substitution ; and as the interchange of § 42 transforms the one into the other, it follows that both are doubly-invariant. (50) The two new forms of eliminant just reached make clear the fact that if one of the four sets of determinants 4,5,6 ; 4', 5', 6' ; 1,2,3 1', 2', 3' vanishes, the eliminant takes the form of a single four-line determinant. For example, if 4, 5, 6 have each the value zero, the eliminant is + 17' 19 ^ 11' H -T 2 i 2 8 °4 27 3 22' 3 38 1 9 39' + T 33' 2 4' 5' 6' + 0' We are thus brought to consider the problem of finding the set of four equations whose coefficients are the elements of this determinant. In the quest for a solution we are not without a lead, since for one of the very special cases brought into notice by Sylvester the desired set of equations has already been obtained. * (51 ) From the fundamental set of equations there can be deduced (§ 33) | u x c 2 h 3 | = -9x 2 + 2y 2 -7'yz+l'xy = 0, j ?/ 1 a 2 & 3 | = 0z 2 + lyz+4zx + 7xy = 0, and from these by multiplication by z and y respectively we obtain two equations in- volving the desirable facients yz 2 , zx 2 , xy 2 , xyz, together with the undesirable y 2 z. On eliminating the last mentioned there results (02 + 17>2 2 +19ac 8 + 27*y 2 + (24 - W)xyz = 0, and by cyclical substitution 38^ 2 + (03 + 28V 2 + 27av/ 2 + (35-22')x?/z = 0, 38 ? /z 2 + 19zx 2 + (01 + 39>(/ 2 + (16-33>2/2 = 0. * Proc. Roy. Soc. Edin., xx. p. 377. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 16). 58 396 From these we obtain DR THOMAS MUIR ON THE (o + ' 7 <y 9z.r 2 + 1 ,; lyz* + (0 + |8')z* 2 + 7xif + U-i l'te = 9 / 2 A y82/« 2 + 9za! 2 + (o + y 9 'W + ( 6 ~ y 3 ') x ^ = ° The performance of the interchange of § 42 gives the companion set 8fz + \lxhj + (0+^9'V* + + J 8' )a 6„ „ . /„ . 6„A« -9z 2 .c + ^2-^5'W = 9z 2 z + (3 - 1 6'W = 7x 2 y + ^8yh + (o+±7'\*z + (l - ^4'W = 0. The necessary fourth equation for the cases alluded to in the preceding paragraph is got from the cubic M 3 of § 9, ix 2 y + byh + 6z 2 x + Myz 1 + b'zx 2 + <o'xif- + (0 + 0')xyz = , or from its companion M' 3 , •2'x 2 y + 3'fz+Vz' i x+2yz 2 + 3zx 2 + lxy 2 + (0 + 0')xyz = 0, by putting three appropriate coefficients equal to zero. (52) Another special case of similar type is still more interesting, viz., the case where 7', 8', 9' vanish. The Jacobian of the given set of equations, viz. -2(8'x 3 ) + 2(24 + 2> 2 2/ + 2(2-2 + i')yz 2 + (4-0 + Q')xyz = 0, then loses three of its terms ; and as the operation 2M 3 + M' 3 gives 2(2-4 + 2> 2 // + 2(2-4' + 2)yz 2 + 3(0 + 0')xyz = 0, it is clear that there follows 2(2-4)v/2 2 + (0-2-0>?/z = 0, — an equation which can be used to complete the first set of three in § 51. Since the vanishing of 7', 8', 9' makes 11', 22', 33' identical, the resulting eliminant is 19 2 ?? 9 7 27 3 4-2' 5-3' 6-1' 2-4' 3-5' 1-6' 0-20' EL1MINANT OF A SET OF GENERAL TERNARY QUADRICS. 397 Had we used the operation M 3 + 2 M' 3 we should have obtained the alternative form 8 59 4 2-4' 67 5 9 3-5' 7 48 6 1 -6' 4-2' 5-3' 6-1' 0--2-0' In this case, however, the two forms are not even superficially different, the one being obtainable from the other bv changing rows into columns and attending to the identities 26 = 89 , 34 = 97 , 15 = 78: or, 159 = 267 = 348 = 789. ( 399 ) XVII. — Theorems relating to a Generalization of Bessel's Function. II. By the Rev. F. H. Jackson, B.N. Communicated by Dr W. Peddie. (MS. received February 6, 1905. Read February 20, 1905. Issued separately April 18, 1905.) § 1. Introduction .... § 2. Function ~E p (x) .... § 3. Expressions for Jacobi's Functions CONTENTS. PAGE 399 § 4. J'„» . . 400 § 5. Various series 403 PAGE 405 407 1. Introduction. The theory of the functions commonly known as q functions might perhaps be greatly developed, if investigators were to work on lines suggested by the functional notation of well-known analytic functions. For instance, the analysis connected with the circular functions sin x , cos x , . . . , might be regarded as the theory of certain infinite products without using any special functional notation. It need not be explained however, how great was the gain to elementary algebra by the introduction of the exponential function (regarded as the limit of a certain infinite product, or as the limit of a certain infinite series) denoted e x , with certain characteristic properties, enabling the worker to make transformations easily and quickly. Of course, the vast store of interesting and in many cases useful results connected with the elementary functions of analysis might have been obtained without the introduction of any •notation capable of rapid and easy transformations, but I think it unlikely that they would have been obtained. In chapter xi. of Cayley's Elliptic Functions the identity 1-g 2 -!-^ . 1 f" ) 1 - q* ' 1 _ r/ 2«+2 T I _ q 2,\ _ q i ■ 1 _ (? 2..+2. 1 _ q 1n+i + ^,<r--ip> 0: is used in order to express Jacobi's function, in the well-known form 1 - 2q cos 2x + 2<? 4 cos 4x - 2q 9 cos Qx + .... The likeness of the series ( I ) to Bessel's series is very obvious. It is a very special case of the series which I have denoted J [n] in previous papers, and in itself might have suggested a theory of q functions analogous to Bessel's functions. In the discussion TRANS. ROY. SOC. EDIN , VOL. XLI. PART II. (NO. 17). 59 400 THE REV. F. H. JACKSON ON of q functions a great variety of notations has been used. I propose in this paper to bring before the Society a series of formulae relating firstly to a function E i ,(x) analogous to exp (x). These formulae are supplementary to those given in Trans. Roy. Soc. Edin., vol. xli. pp. 105-118, and will lead to one or two interesting properties of a function J p m , n (x), which may be termed a generalized Bessel-function of double order, and to various novel expressions of elliptic functions in terms of the generalized Bessel- function. For example, Jacobi's O function is expressed by the form • (A) in which *( 2K«\ 7T / _ q I[»]( M )I[» 2o = fi (1 - q~ m ) m=l W = "Fl V = = » Jv P -i* FT n = K' It is noteworthy that n (the order of the J functions) is in (A) an arbitrary number. It appears only in the expression on the right side of that equation. A definite integral expression for the functions J will also be given. 2. Function E p (x) . The series and its equivalent product 1+— +p + P — 1 p - l-p 2 - \ (l-x)(l-pz)(l-p*x) are well known : we derive a function analogous to the exponential function. (Cf. Trans. R.S.E., xli. p. 116; and Proc. L.M.S., series 2, vol. ii. p. 194.) iT'W kr ■E 1 (x) = l+£-+p r ^ r + .... +.p*-vpz + X . X* , X [r] = (p r -l)f(p-l) The function E p (x) may be regarded, like the exponential function, either as the limit of a certain infinite series or a certain infinite product. The results numbered (2) . . . (26) are either easily obtained or are known in other forms. THEOREMS RELATING TO A GENERALIZATION OF BESSEL'S FUNCTION. 401 E»-E L (-x) = l (2) p E»-E,( -*) - I + -^ (1 - P y> +T -^L— t (1 -?)< + (3) -"*(".£l) w which reduce, when p= 1, to exp (x) x exp (x)=l, E.(*)-E.(-J!LJ) = l + * a+ ^ t 7 + (5) p \p-iJ "V p-v i-p 2 \-p 2 -l-p* ,.2 = 5(1 -a^p-"") • • ■ • • ( 7 ) m=l The product is absolutely convergent if | p | > 1. The series are convergent, however, P if | p |^1, and also for | /> j < 1 provided x <-^ It follows that if \p\ > 1, «, = (-!)- E^^E^ - o^E^oAe^ _ ^ E i) (co^)E J) (-o>"x) = E ;)2 „^» ( -^ r l^ n ) . . (8) The corresponding theorem in case p <1 is easily obtained by inversion of the base_p. = n [\-xY m \ 0°) m=l ( ) Ei/^^y E L f - ^S) = n 1 1 --! rv m_1 !• • • • • • (U) p\»-l/ v \ P—IJ m=l I J = ? i> v-l)(p*-l)...(p a "-l) • • ■ (13) • ■ ... (H) n^J • E ^f3p2 J = ^ (1 aM) - i - 2 p(* 2 + *~ 2 ) + 'M* 4 + *-') - 2PV + *~ 6 ) + m=l On putting x = e™, the series on the right becomes Jacobi's function e( 2K " in which -ji-4^ J9 = e K E^fr) - 1 + <!!!+* + (' + gj^^ + ■ • • • (15) »<1, .t< , y unrestricted l-p (Cf. Proc. Edm. Math. Soc, vol. xxh. (8).) or p>l, y< , x unrestricted. p-l 402 THE REV. F. H. JACKSON ON From this we derive ^h(A)- 1+w ' +M PTr ] ^ + <16) -/>> <i7) *GqMC£i)- I+ ' + " + -"- • • <18 > • V' j \p ■ l l-x (x<l) . . (19) In this expression we notice that inversion of the base p simply interchanges the E functions in the product on the left side of the equation (18). E ^H~) - 1 - [»> + 1^* - P lnIn ~$ l - 21 * + (20) = #("*> (21) Hence The equations are special cases of (25). If n be infinite f\x) = f (v n 'A (22) (z)x/ (-a:)-/ (x 2 ) (23) ^IW^K-^^ 2 ) ... . (24) _/%)x<K(-*) = l (25) e* • e - * = 1 E».E i (-x) = l />>= e v(^t) ■'.••■.• (36) Function I [n] (.x) . It is well known in the theory of Bessel's function that -* t t \ - x " i i , 2« + 3 2 2u + 5 3 I € Pl * W ~2-.r(fi+l)1 !C + 2(2» + 2) a: ' 2 3(2n + 2) a! + J I„(a;) = t- n J B (» a! ). In a paper on Basic numbers applied to Bessel's function (Proc. Loud. Math. Soc, series 2, vol. iii., 1905), I have extended this theorem in the form THEOREMS RELATING TO A GENERALIZATION OF BESSEL'S FUNCTION. 403 E p (-x)I m (x) = E,(-s)I [n] (z) ; " l.l- x + [ 2 » + 3 ]^ 2_ [2» + 5] x5 I (27) {2m}! 1 [2][2» + 2] [2][3][2» + 2] ![»](*) = *~"J[»](*«) /[„](*) = *-"?[„](**) the conditions for convergence being as follows : Case i | p | > 1 E p (x - ) and I [n] (a;) are absolutely convergent for all values of x. Ei(«) and / [n] (#) are absolutely convergent if x< * p p — l Case ii | p | < 1 Ej(a;) and l[ ni (x) are absolutely convergent for all values of x. p E p (x) and I w (x) are absolutely convergent if x< • \-p The series (27) is convergent for all values of p. It is easily deduced that l [n] (;x) = -E p (ix)E p (-ix)% in ix) (28) | rn] (.T) = E J (^)E 1 (-^)J [n] (x) (29) p p Prom these relations some interesting expressions for various elliptic functions may be found. 3. Relations with Elliptic Functions. By means of equation (29) we are able to write Iw^OIw^^) = E L (tirf)E L ( - ixt)EiJixt-^)}L L { - ixl-i) . . . (30) "["]( a 'V"[>'](' C ^ ) P P P P Replacing x by u, (v = ixjpjip — 1)), we obtain by means of result (11) Iw KJImC"*" 1 ) = n ((i-^y-')(i-A-y"-') I . . . (31) Using result (12), the right side of this equation may be written or = E 1 ( r 1 .^)xE 1 f r V^) .... (33) 404 THE REV. F. H. JACKSON ON This expression, when expanded in a Laurent series of ascending and descending powers of t, takes the form (Trans. R.S.E., vol. xli. p. 117 (m)), / i'(^) + Z ( - 1 ) m ^ 2m+r2m ^ m2/ '"'(^i) • • • (34) In case x = 1, the product II ( (1 -xHY m ~ l ){l -xH^p^' 1 ) ) may be expressed as I.- n(i-_p2»») m=l ' p (^2 + ^-2) + ^4 + r 4) _ , _ . (35) (Cf. Cayley's Elliptic Functions, p. 297, ed. 1876.) We see incidentally that 1 1 n(i-jo 2m ) l V-i •.x-Wr^ ( 36 ) m— 1 for all positive integral values of n. Denoting the nature of the base by an index, we write #-\) m \p - 1) m \p - 1 ^L4T)=^t L r) = ^(A) = (37) I'd— W 11 ^— )=^ 4li (r^) = < 38 > which is the expression in generalized Bessel-function-notation of the well-known result 1 I 2n+2 I °° 1 1 - ? 2 • 1 - 2 * ... 1 - q 2m I (1 - ? 2 )(1 - g 2n+2 ) I m-i(l - q 2m ) On replacing t by <^ 9 , the equation (31) becomes 19 = n(l -2x 2 cos2fe 2m - 1 + a;V m " 2 ) • • (39) -i9\ Using now Jacobi's notation, and writing u = *P e ix ,v = *P e' ix q = U(l-q 2m ) p-l p-l '* m = l * q — p, we obtain q M^) q(MA (40) J m( M )J[»](^) V TT ) K ' 2^smx.f ] (« n )W = H /2K ? \ _ Jw(^ l «)J[»](l» J t») V 7T / V ' 2gi cos x IE . SMMU&WMnUifto) _ JZKx\ u „ j¥ ^b^\¥rb^ - 4—) .... (44) THEOREMS RELATING TO A GENERALIZATION OF BESSEL'S FUNCTION. 405 We notice that in the expressions for sn, en, dn, two arbitrary constants (orders of the functions) n, m appear : Jacobi's function Z(x) = , / is related to the J functions as follows, 2K(^1 ) . y rlK^ = pix i |» _ J» I _ prix < f » _ Jjy) 1 ^ v - y ' ![»](«) Jw(«) ' < Iw(») JwW ' It is plain that Weierstrass's functions <r, £, p, may be expressed by similar formulae. For convenience of printing, the order of the functions will sometimes be expressed by n instead of [n]. The known formulae of Jacobi's functions will, it is evident, give rise to corresponding forms in the case of the J functions : for example sn 2 + en 2 = 1 gives rise to jl(P l «)Jl(P i «) Jl(*>>*«)Jl(V J t;) WfMJHv) (46) = u & e" , v = — ^-e ix ,p = q = e n k • jj - 1 ^ - 1 Using (11) and (12) it is easily found by the method of § 10, p. 1 16, vol. xli., Trans. R.S.E., that jl { 1 - Sap*"- 1 cos x + ay-' J = ^(-^l) + %JjT cos nxlj^-^) . By Fourier's theorem we write therefore P^-Z-Tt) = <T~ { n ( X ~ 2a P 2m-1 cos * + aV"" 2 ) I cos nz • dx . . (47) "Jo In case a = 1 , this reduces to «-<^i)=iJ o cos ^- e (v>- • ■ < 48 > 4. Function J p (x) . n,m\ I Forming in a series, according to powers of t , the product J M (xt) x Jp^arf" 1 ) we obtain in which J^(*) + 2(-l) m (< am + <- ,m )0*) • ( 49 ) n ' m X ~ fr '{2m + 2r* + 2r}!{2m + 2r}\{2n + 2r}!{2?'} ! {2r}! = [2][4] .... [2r] 406 THE REV. F. H. JACKSON ON In the same way if we take ? ,( ' C) = ^ ( " 1)r {2m + 2n + 2r}!{27n + 2r}!{27 l + 2r}!{2r}! / ' 4r<m+ " +rl ' " (51) which is related to J nm by inversion of the base p, since J? (x) = p 2{m3+ "' +mn) p> (x) ,,,m -"- mX Taking the product of two J series we find ! l „3(*0><I[»j(^- 1 ) = S^ (^) + 2(- 1 )V m(m+n, (^ m + ^ i!m )I* m (a;) . . . (52) In a previous paper (Trans. R.S.E., vol. xli. p. 106) it has been shown that j,„ )W ) x i,„w~ ± < - n {M j^^ { \ t +2rmr) r -+" ■ • p»» There is a certain similarity of form among the series (50), (51), (53). Consider now the product of four J functions J M (arf) • J m (xt) ■ Mxt- 1 ) ■ f^arf" 1 ) .... (54) This expression may be written in two other forms. Firstly, by (49) and (52) we write it Secondly, by means of (53) we express (54) as 1 V ( _ iy {2n + 2v + 4r}! ( x t) m+ * +2r I • I fbi / {2« + 2v+2r}![2ra + 2r}!{2v + 2r}!{2r}P ' ) If/ iy {2w + 2v + 4r}! , f _ 1)n+v+2r ) . l£tf V ' {2n + 2r + 2v}\{2n+2r}l{2v + 2r}\{2r}\ K ' /•■• • V») Equating coetiicients of powers of t in (55) and (56), we find from the terms independent of t y( {2n+2v + ir}\ \ 2 x 2 "+-'+ tr = J P P +2Y p 2mim+ ^J p P . (57) ^rf,\{2n + 2v+2r}\{2n + 2r}l{2v+2r}l{2r}\J "•"-■° ^ "•»-*» The terms in the series on the left side of (57) are the squares of the terms in (53). Generally y {2n + 2v + ir} \{2n + 2v + 4???. + 4r} ! a ; 2 "' +2 ' ,+2 '' +4 '- ^{2wi + 2n + 2v + 2r}!]27» + 2v + 27-}!{2m + 2»4-2r}!{2ra + 2v + 2r}!{2m + 2r}!{2n + 2r}!{2v + 2r}!{2r}! _ "V n'imim+p) j J 1 ' %" . n *mr+2r(r+n) T* 9 P I .... (58) r=l ' ' THEOREMS RELATING TO A GENERALIZATION OF BESSEL'S FUNCTION. 407 5. In this section of the paper I propose to state briefly some results which may be deduced by means of (53), (28), (29), \L p (ix)T£ p ( - ix) r»2 /y.4 = 1 - ^ ^< 1 -^ + ^^V-^ 1 "" ), '" = J m {px)JU x ) + HiW^i^) + 2 P iJ m(P x ) J m( x ) + Indicating the nature of the base of each function by an index, we write V ( x \ i\ p2 ( * 2 \ + 2 V Di p2 V - % v ( X whence by (29) ■p ( x l) J S(ynri) + 2 2 J p»o • <C - H^ri) E A " ^i)K(^ri by (29) by (4) and 12) -^ {p * +1)sr ^~ (1 +p)S , ^. From (19) we find (l-xH-*)(\-zW) ■Jp 'H£W£i 1 -cc* 1 1 ~ 2cc 2 cos 20 + .,- 4 I | I [0 j+2cos2^I m + . . . I J J m - 2pcos20/ [1] + . . . 1 I ^4 2= [T] l[0]|ra+ 4T] lco]l[3]+ +P rlT - 1 %-»&-v ■ ■ E p (ix)E p ( - ix) = { J„ } ' + W { J U] | 2 + . . . . + W^j { J w } 2 + . TRANS. ROY. SOC. EDIN, VOL. XLI. PART II. (NO. 17). (59) (60) (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) 60 408 THEOREMS RELATING TO A GENERALIZATION OF BESSEL'S FUNCTION. W 2 -M J "! 2+ HM 2 - ={Z(-^ (1 _ p2)(1 .X..(i-W y " ir } {z (1 ^^^:: a (1 ?r ( - 1)2 1 (71) W 2 -[1]N} 2 + +^f{M 2 ---- = { 2(- i )' (1 _^ )(1 _ i ^ . . . (i^^p- 1 ) 1 - 1 / V 2(l+p2)(l_ p 4 ) (1 +^-0 ■ (1 +j? 2 )(l +^) . . (1 + j^) _ 1)9r I , 72 v 1 ^~ {(1-pW^V^ • (i -p 2r )\ 2 ^ < K J (Cf. Trans. #.£.#, p. 110 (26).) J[o](*)J[o](2/) - 2 Mi](^)J[n(2/) + 2p 4 J [2] (a;)J [2] (y) - . . . . ""I* fT^ x > + 1-^2.1-74^ *> -"M [2]» + [2]»[4]« J ^ V^ + W t ^- ^«-*J) 1 [2] ^ +[4F M +[8]J ^ + I • • (74) X' n ~ l [4:71 + 21 E p (ix)E p ( - ix ) c 2n y< 2n _ 2 y = J[»-i]J M + ■ [2] J[ " ]J[ « +1] S ( S -du , -< - 1 [4ra + 2][4?i + 4] [4« + 2s-2] 7 T n ~. +p s(° i>[4 re + 4s - 2] L JL j ^ ^ l _ JJ [m+s] J [m+s _ 1] (75) and a similar form for x 2n . (Cf. Proc. Lond. Math. Soc, series 2, vol. iii.) ^ ( } ~FT P [^]"[2] s -r^)1 ^ ( - !) (i -j>")(i-j>*) • . . (i-W p_1) f ( ^2(1 +F 8 ) • • (l+^ 2 '- 2 )-(l+p) . . . (1 +/'- 1 ) T ,,- 1 1 (7g) (Cf. Proc. Edin. Math. Soc, Theorem of Lommel, vol. xxii.) It is plain that great numbers of such theorems may be found and expressed in various forms by means of the transformations belonging to E^a:)*, but the examples given above will suffice to illustrate the notation. ( 409 ) XVIII. — On Pennella balaenopterae : a Crustacean, parasitic on a Finner Whale, Balasnoptera rnusculus. By Sir William Turner, K.C.B., D.C.L., F.R.S. (With Four Plates.) (Read February 6, MS. received February 8, 1905. Issued separately May 26, 1905.) CONTENTS. Introduction 409 External Characters of the Female . . 412 Chitinous Envelope 414 Structure op Head 414 Alimentary Canal 419 Nervous System . . . . • . . 422 Pennate Appendages 423 Reproductive Organs 424 The Male 427 Comparison with other Species . . . 428 conchoderma 430 Bibliography 431 Explanation of Plates 432 Introduction. In September 1903 I received a bottle containing twelve specimens of a large parasite presented to me by Mr Chr. Castberg, the manager of a Norwegian whaling company which has established a fishing station at Ronasvoe in the north of Shetland.* In his letters Mr Castberg stated that the parasites were attached to a Finner whale, which, from its size, the mottled character of the whalebone and the pointed head, was obviously a Razorback — Balsenoptera rnusculus. The parasites were numerous, and were fixed to the back of the whale, and the attached end penetrated through the skin into the blubber. Although Mr Castberg had seen many hundred whales, this is the first occasion on which he had met with this form of parasite. From the characters of the specimens I concluded that they were a giant species of a parasitic Crustacean, of the family Lernseidse, and on further investigation I associated them with the genus Pennella (Oken). This genus is now regarded as including those members of the Lernseidse which, as studied in the females, have the head stunted and club-shaped, with horn-like arms radiating from its base ; the body elongated, cylindriform, not bent into a sigmoid shape ; the anterior part of the body attenuated, but widening further back ; a pair of genital openings with depending ova strings ; the terminal part of the body caudate, giving origin to the characteristic bristle-like pennate appendages ; pairs of minute rudimentary feet springing from the ventral surface of the body close to the base of the head. From the time of Aristotle, naturalists had recognised that the Tunny and Swordfish were infested by worm-like parasites, fastened to the skin near the fin. Rondeletius, * I am indebted to my valued correspondent, Mr Thomas Anderson, merchant, of Hillswick, Shetland, for putting me into communication with Mr Castberg. TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 18). 61 410 SIR WILLIAM TURNER Gesner and Salvianus, in their respective treatises, written in the sixteenth century, described such parasites, and Rondeletius and Gesner figured specimens from the tunny. Boccone published in 1674 an account of parasites found on the swordfish, Xiphias, implanted in its flesh, which he named Sangsue or " Hirudo cauda utrinque pinnata," and he gave a figure. It would seem as if this animal was different from that described by Rondeletius and Gesner. Boccone had figured a very interesting object, named by him a " poux " or " pediculus," as big as a pea, attached to the ventral surface of the parasite, immediately in front of the genital openings. He stated that it was fixed as firmly to the parasite as a limpet was to a rock. I am disposed to regard this so-called " pediculus " as the male of the female parasite to which it was attached. Its small size compared with that of the female, and its position and attachment close to the genital openings, corresponded with that of the male of the parasitic crustacean, Chondra- canthus lophii, described and figured by Dr H. S. Wilson and myself in 1862. Linnaeus, in the Sy sterna Naturw, 1758, classed amongst the Vermes Zoophyta the genus Pennatula or Sea Pens, and he named the parasite described by Boccone, which infests Xiphias, JPennatula Jilosa. In 1759 J. L. Odhelius contributed to the Amceni- tates Academics of Linnaeus, a dissertation entitled " Chinensia Lagerstromiana," # in which he gave the characters of Pennatula sagitta (p. 257, and fig. 13), a parasite infesting Lophius Iristrio, the sea-bat of the China Sea. John Ellis reproduced in 1764 Boccone's figure of P. Jilosa and Odhelius's figure of P. sagitta. In 1802 Holten recognised a parasite on the flying fish, Exocsetus volitans, which he named P. exocseti, specimens of which, burrowing into the abdominal cavity of that fish, have been recently described, 1901, by Mr Andrew Scott. Oken classed the Lernaeidse amongst the Mollusca, removed these parasites from the Sea Pens, Pennatula, and placed them in a distinct genus, Pennella, whilst De Blainville suggested Lerneopenna as the generic name. Cuvier and naturalists generally had adopted Oken's term, though some preferred the spelling Penella. Additional species were discovered from time to time. Chamisso and Eysenhardt described Penella diodontis from the branchiae of Diodontis mola, captured in the Pacific ; Dekay named P. sagitta as adhering to Diodon pilosus, and von Nordmann, in his description of P. sagitta from Lophius marmoratus, thought that it and Dekay's specimen were the same species. Angus found a parasite on a species of Coryphwna near the gills, which William Baird named Penella pustulosa. Milne Edwards stated that Pennella sultana had been found in the mouth of Carenx ascensofius. Steenstrup and Lutken gave an account of P. varians which infested a "Dolphin," the species of which was not determined. E. Perceval Wright described Pennella orthagorisci from specimens obtained from Orthagoriscus mola caught in Cork har- bour in 1869. They were implanted in the skin on either side of the dorsal fin, and the total length of the parasite from the head to the anal opening was 7 inches. He * Named after the Swedish Councillor, Magnus Lagerstrom. ON PEN NELL A BALDEN OPT ERA:. 411 also stated that Baird referred a Pennella from a sunfish captured in Cornwall to P. filosa. G. M. Thomson gave an account (1889) of a Pennella found on a swordfish (Histiophorus herschelii), which he named P. histiophori. Eamsay H. Traquair has called my attention .to two specimens of Pennella in the Collection of the Royal Scottish Museum, which he had provisionally named P. orihagorisci. Possibly they may have been included in the Natural History Museum of the University, which was transferred many years ago to the Royal Scottish Museum, but nothing definite is known of the animal on which they were parasitic, or when they were obtained. One specimen was deprived of the head and arms ; the other had a head and two lateral arms, but no dorsal arm, and it was about 5 inches long.* Observations on the Lernaeidae during the first quarter of the last century induced naturalists to consider that these parasites were not to be regarded as Worms, Molluscs, or Zoophytes, but that they had an affinity to the Crustacea. Their position was finally adjusted in 1832 by Alexander von Nordmann, who, from the young having the non-parasitic character of Cyclops, from the segmented structure of the male, which is a free swimming animal, though it may become attached to the female, and from the position and characters of the feet, definitely placed these curious animals amongst the Crustacea, in which they are now generally regarded by naturalists as forming a family of parasitic Copepoda. An important extension of our knowledge of the hosts to which different species of Pennella may become attached was made when it was ascertained that specimens had been obtained imbedded in the skin of species of whales frequenting the North Atlantic Ocean. Steenstrtjp and Lutken published in 1861 a memoir in which a Pennella was described as attached to a Hyperoodon rostratus captured in 1855 south of the Faroe Islands ; they named the parasite Pennella crassicornis. They referred to an observation made some years previously by von Duben that a Pennella, species not named, had been obtained from a Finner whale. In 1866 G. 0. Sars stated that specimens of a Pennella with the head buried in the blubber were seen attached to Balsenoptera musculus. In 1877 Koren and Danielssen published a memoir on a Pennella found on Balsenoptera rostrata, and preserved in the museum at Bergen, which they had named Pennella balsenopterse twenty years previously. Other specimens from B. rostrata, buried with the head and horn-like arms in the blubber in the vicinity of the external organs of generation, had subsequently been added to this museum. Van Beneden, in his memoirs on the natural history of the Cetacea, referred to these Balgenopterae as serving as hosts for a Pennella ; and he further stated, though without giving very definite authoriti